id	sid	tid	token	lemma	pos
ejpam-4396	1	1	european	european	PROPN
ejpam-4396	1	2	journal	journal	PROPN
ejpam-4396	1	3	of	of	ADP
ejpam-4396	1	4	pure	pure	ADJ
ejpam-4396	1	5	and	and	CCONJ
ejpam-4396	1	6	applied	apply	VERB
ejpam-4396	1	7	mathematics	mathematic	NOUN
ejpam-4396	1	8	vol	vol	NOUN
ejpam-4396	1	9	.	.	PROPN
ejpam-4396	2	1	15	15	NUM
ejpam-4396	2	2	,	,	PUNCT
ejpam-4396	2	3	no	no	INTJ
ejpam-4396	2	4	.	.	NOUN
ejpam-4396	2	5	3	3	NUM
ejpam-4396	2	6	,	,	PUNCT
ejpam-4396	2	7	2022	2022	NUM
ejpam-4396	2	8	,	,	PUNCT
ejpam-4396	2	9	887	887	NUM
ejpam-4396	2	10	-	-	SYM
ejpam-4396	2	11	896	896	NUM
ejpam-4396	2	12	issn	issn	PROPN
ejpam-4396	2	13	1307	1307	NUM
ejpam-4396	2	14	-	-	SYM
ejpam-4396	2	15	5543	5543	NUM
ejpam-4396	2	16	–	–	PUNCT
ejpam-4396	2	17	ejpam.com	ejpam.com	X
ejpam-4396	2	18	published	publish	VERB
ejpam-4396	2	19	by	by	ADP
ejpam-4396	2	20	new	new	PROPN
ejpam-4396	2	21	york	york	PROPN
ejpam-4396	2	22	business	business	PROPN
ejpam-4396	2	23	global	global	VERB
ejpam-4396	2	24	some	some	DET
ejpam-4396	2	25	properties	property	NOUN
ejpam-4396	2	26	of	of	ADP
ejpam-4396	2	27	g	g	NOUN
ejpam-4396	2	28	-	-	PUNCT
ejpam-4396	2	29	groups	group	NOUN
ejpam-4396	2	30	joey	joey	PROPN
ejpam-4396	2	31	a.	a.	PROPN
ejpam-4396	2	32	caraquil1	caraquil1	PROPN
ejpam-4396	2	33	,	,	PUNCT
ejpam-4396	2	34	michael	michael	PROPN
ejpam-4396	2	35	p.	p.	PROPN
ejpam-4396	2	36	baldado	baldado	PROPN
ejpam-4396	3	1	jr.2,∗	jr.2,∗	PROPN
ejpam-4396	3	2	1	1	NUM
ejpam-4396	3	3	southern	southern	ADJ
ejpam-4396	3	4	leyte	leyte	PROPN
ejpam-4396	3	5	state	state	PROPN
ejpam-4396	3	6	university	university	PROPN
ejpam-4396	3	7	,	,	PUNCT
ejpam-4396	3	8	tomas	tomas	PROPN
ejpam-4396	3	9	oppus	oppus	PROPN
ejpam-4396	3	10	,	,	PUNCT
ejpam-4396	3	11	southern	southern	ADJ
ejpam-4396	3	12	leyte	leyte	PROPN
ejpam-4396	3	13	,	,	PUNCT
ejpam-4396	3	14	philipppines	philipppine	NOUN
ejpam-4396	3	15	2	2	NUM
ejpam-4396	3	16	mathematics	mathematics	NOUN
ejpam-4396	3	17	department	department	NOUN
ejpam-4396	3	18	,	,	PUNCT
ejpam-4396	3	19	negros	negros	PROPN
ejpam-4396	3	20	oriental	oriental	ADJ
ejpam-4396	3	21	state	state	PROPN
ejpam-4396	3	22	university	university	PROPN
ejpam-4396	3	23	,	,	PUNCT
ejpam-4396	3	24	dumaguete	dumaguete	PROPN
ejpam-4396	3	25	city	city	PROPN
ejpam-4396	3	26	,	,	PUNCT
ejpam-4396	3	27	philippines	philippine	NOUN
ejpam-4396	3	28	abstract	abstract	ADJ
ejpam-4396	3	29	.	.	PUNCT
ejpam-4396	4	1	a	a	DET
ejpam-4396	4	2	nonempty	nonempty	ADV
ejpam-4396	4	3	set	set	VERB
ejpam-4396	4	4	g	g	NOUN
ejpam-4396	4	5	is	be	AUX
ejpam-4396	4	6	a	a	DET
ejpam-4396	4	7	g	g	NOUN
ejpam-4396	4	8	-	-	PUNCT
ejpam-4396	4	9	group	group	NOUN
ejpam-4396	5	1	[	[	X
ejpam-4396	5	2	with	with	ADP
ejpam-4396	5	3	respect	respect	NOUN
ejpam-4396	5	4	to	to	ADP
ejpam-4396	5	5	a	a	DET
ejpam-4396	5	6	binary	binary	ADJ
ejpam-4396	5	7	operation	operation	NOUN
ejpam-4396	5	8	∗	∗	NOUN
ejpam-4396	5	9	]	]	PUNCT
ejpam-4396	5	10	if	if	SCONJ
ejpam-4396	5	11	it	it	PRON
ejpam-4396	5	12	satisfies	satisfy	VERB
ejpam-4396	5	13	the	the	DET
ejpam-4396	5	14	following	follow	VERB
ejpam-4396	5	15	properties	property	NOUN
ejpam-4396	5	16	:	:	PUNCT
ejpam-4396	5	17	(	(	PUNCT
ejpam-4396	5	18	g1	g1	PROPN
ejpam-4396	5	19	)	)	PUNCT
ejpam-4396	5	20	a	a	DET
ejpam-4396	5	21	∗	∗	NOUN
ejpam-4396	5	22	(	(	PUNCT
ejpam-4396	5	23	b	b	NOUN
ejpam-4396	5	24	∗	∗	NOUN
ejpam-4396	5	25	c	c	NOUN
ejpam-4396	5	26	)	)	PUNCT
ejpam-4396	5	27	=	=	NOUN
ejpam-4396	5	28	(	(	PUNCT
ejpam-4396	5	29	a	a	DET
ejpam-4396	5	30	∗	∗	NOUN
ejpam-4396	5	31	b	b	NOUN
ejpam-4396	5	32	)	)	PUNCT
ejpam-4396	5	33	∗	∗	NOUN
ejpam-4396	5	34	c	c	NOUN
ejpam-4396	5	35	for	for	ADP
ejpam-4396	5	36	all	all	DET
ejpam-4396	5	37	a	a	DET
ejpam-4396	5	38	,	,	PUNCT
ejpam-4396	5	39	b	b	NOUN
ejpam-4396	5	40	,	,	PUNCT
ejpam-4396	5	41	c	c	PROPN
ejpam-4396	5	42	∈	∈	PROPN
ejpam-4396	5	43	g	g	NOUN
ejpam-4396	5	44	;	;	PUNCT
ejpam-4396	5	45	(	(	PUNCT
ejpam-4396	5	46	g2	g2	PROPN
ejpam-4396	5	47	)	)	PUNCT
ejpam-4396	5	48	for	for	ADP
ejpam-4396	5	49	each	each	PRON
ejpam-4396	5	50	a	a	DET
ejpam-4396	5	51	∈	∈	PROPN
ejpam-4396	5	52	g	g	NOUN
ejpam-4396	5	53	,	,	PUNCT
ejpam-4396	5	54	there	there	PRON
ejpam-4396	5	55	exists	exist	VERB
ejpam-4396	5	56	an	an	DET
ejpam-4396	5	57	element	element	NOUN
ejpam-4396	5	58	e	e	PROPN
ejpam-4396	5	59	∈	∈	PROPN
ejpam-4396	5	60	g	g	PROPN
ejpam-4396	5	61	such	such	DET
ejpam-4396	5	62	that	that	SCONJ
ejpam-4396	5	63	a	a	DET
ejpam-4396	5	64	∗	∗	NOUN
ejpam-4396	5	65	e	e	NOUN
ejpam-4396	5	66	=	=	NOUN
ejpam-4396	5	67	a	a	DET
ejpam-4396	5	68	=	=	SYM
ejpam-4396	5	69	e	e	NOUN
ejpam-4396	5	70	∗	∗	NOUN
ejpam-4396	5	71	a	a	PRON
ejpam-4396	5	72	(	(	PUNCT
ejpam-4396	5	73	e	e	NOUN
ejpam-4396	5	74	is	be	AUX
ejpam-4396	5	75	called	call	VERB
ejpam-4396	5	76	an	an	DET
ejpam-4396	5	77	identity	identity	NOUN
ejpam-4396	5	78	element	element	NOUN
ejpam-4396	5	79	of	of	ADP
ejpam-4396	5	80	a	a	PRON
ejpam-4396	5	81	)	)	PUNCT
ejpam-4396	5	82	;	;	PUNCT
ejpam-4396	5	83	and	and	CCONJ
ejpam-4396	5	84	,	,	PUNCT
ejpam-4396	5	85	(	(	PUNCT
ejpam-4396	5	86	g3	g3	NOUN
ejpam-4396	5	87	)	)	PUNCT
ejpam-4396	5	88	for	for	ADP
ejpam-4396	5	89	each	each	PRON
ejpam-4396	5	90	a	a	DET
ejpam-4396	5	91	∈	∈	PROPN
ejpam-4396	5	92	g	g	NOUN
ejpam-4396	5	93	,	,	PUNCT
ejpam-4396	5	94	there	there	PRON
ejpam-4396	5	95	exists	exist	VERB
ejpam-4396	5	96	an	an	DET
ejpam-4396	5	97	element	element	NOUN
ejpam-4396	5	98	b	b	PROPN
ejpam-4396	5	99	∈	∈	PROPN
ejpam-4396	5	100	g	g	NOUN
ejpam-4396	5	101	such	such	DET
ejpam-4396	5	102	that	that	SCONJ
ejpam-4396	5	103	a	a	DET
ejpam-4396	5	104	∗	∗	NOUN
ejpam-4396	5	105	b	b	NOUN
ejpam-4396	5	106	=	=	SYM
ejpam-4396	5	107	e	e	NOUN
ejpam-4396	5	108	=	=	SYM
ejpam-4396	5	109	b	b	PROPN
ejpam-4396	5	110	∗	∗	NOUN
ejpam-4396	5	111	a	a	PRON
ejpam-4396	5	112	for	for	ADP
ejpam-4396	5	113	some	some	DET
ejpam-4396	5	114	identity	identity	NOUN
ejpam-4396	5	115	element	element	NOUN
ejpam-4396	5	116	e	e	PROPN
ejpam-4396	5	117	of	of	ADP
ejpam-4396	5	118	a.	a.	NOUN
ejpam-4396	5	119	in	in	ADP
ejpam-4396	5	120	this	this	DET
ejpam-4396	5	121	study	study	NOUN
ejpam-4396	5	122	,	,	PUNCT
ejpam-4396	5	123	we	we	PRON
ejpam-4396	5	124	gave	give	VERB
ejpam-4396	5	125	some	some	DET
ejpam-4396	5	126	important	important	ADJ
ejpam-4396	5	127	properties	property	NOUN
ejpam-4396	5	128	of	of	ADP
ejpam-4396	5	129	g	g	NOUN
ejpam-4396	5	130	-	-	PUNCT
ejpam-4396	5	131	subgroups	subgroup	NOUN
ejpam-4396	5	132	,	,	PUNCT
ejpam-4396	5	133	homomorphism	homomorphism	NOUN
ejpam-4396	5	134	of	of	ADP
ejpam-4396	5	135	g	g	NOUN
ejpam-4396	5	136	-	-	PUNCT
ejpam-4396	5	137	groups	group	NOUN
ejpam-4396	5	138	,	,	PUNCT
ejpam-4396	5	139	and	and	CCONJ
ejpam-4396	5	140	the	the	DET
ejpam-4396	5	141	zero	zero	NUM
ejpam-4396	5	142	element	element	NOUN
ejpam-4396	5	143	.	.	PUNCT
ejpam-4396	6	1	we	we	PRON
ejpam-4396	6	2	also	also	ADV
ejpam-4396	6	3	presented	present	VERB
ejpam-4396	6	4	a	a	DET
ejpam-4396	6	5	couple	couple	NOUN
ejpam-4396	6	6	of	of	ADP
ejpam-4396	6	7	ways	way	NOUN
ejpam-4396	6	8	to	to	PART
ejpam-4396	6	9	construct	construct	VERB
ejpam-4396	6	10	g	g	NOUN
ejpam-4396	6	11	-	-	PUNCT
ejpam-4396	6	12	groups	group	NOUN
ejpam-4396	6	13	and	and	CCONJ
ejpam-4396	6	14	g	g	NOUN
ejpam-4396	6	15	-	-	PUNCT
ejpam-4396	6	16	subgroups	subgroup	NOUN
ejpam-4396	6	17	.	.	PUNCT
ejpam-4396	7	1	2020	2020	NUM
ejpam-4396	7	2	mathematics	mathematic	NOUN
ejpam-4396	7	3	subject	subject	NOUN
ejpam-4396	7	4	classifications	classification	NOUN
ejpam-4396	7	5	:	:	PUNCT
ejpam-4396	7	6	08a05	08a05	NUM
ejpam-4396	7	7	key	key	ADJ
ejpam-4396	7	8	words	word	NOUN
ejpam-4396	7	9	and	and	CCONJ
ejpam-4396	7	10	phrases	phrase	NOUN
ejpam-4396	7	11	:	:	PUNCT
ejpam-4396	7	12	g	g	NOUN
ejpam-4396	7	13	-	-	PUNCT
ejpam-4396	7	14	group	group	NOUN
ejpam-4396	7	15	,	,	PUNCT
ejpam-4396	7	16	g	g	NOUN
ejpam-4396	7	17	-	-	PUNCT
ejpam-4396	7	18	subgroup	subgroup	NOUN
ejpam-4396	7	19	,	,	PUNCT
ejpam-4396	7	20	group	group	NOUN
ejpam-4396	7	21	,	,	PUNCT
ejpam-4396	7	22	homomorphism	homomorphism	NOUN
ejpam-4396	7	23	,	,	PUNCT
ejpam-4396	7	24	zero	zero	NUM
ejpam-4396	7	25	element	element	NOUN
ejpam-4396	7	26	1	1	NUM
ejpam-4396	7	27	.	.	PUNCT
ejpam-4396	8	1	introduction	introduction	NOUN
ejpam-4396	8	2	a	a	DET
ejpam-4396	8	3	binary	binary	ADJ
ejpam-4396	8	4	operation	operation	NOUN
ejpam-4396	8	5	∗	∗	NOUN
ejpam-4396	8	6	on	on	ADP
ejpam-4396	8	7	a	a	DET
ejpam-4396	8	8	set	set	NOUN
ejpam-4396	8	9	g	g	NOUN
ejpam-4396	8	10	is	be	AUX
ejpam-4396	8	11	a	a	DET
ejpam-4396	8	12	function	function	NOUN
ejpam-4396	8	13	from	from	ADP
ejpam-4396	8	14	g	g	PROPN
ejpam-4396	8	15	×	×	PROPN
ejpam-4396	8	16	g	g	NOUN
ejpam-4396	8	17	to	to	PART
ejpam-4396	8	18	g.	g.	VERB
ejpam-4396	8	19	the	the	DET
ejpam-4396	8	20	image	image	NOUN
ejpam-4396	8	21	of	of	ADP
ejpam-4396	8	22	(	(	PUNCT
ejpam-4396	8	23	a	a	DET
ejpam-4396	8	24	,	,	PUNCT
ejpam-4396	8	25	b	b	NOUN
ejpam-4396	8	26	)	)	PUNCT
ejpam-4396	8	27	under	under	ADP
ejpam-4396	8	28	∗	∗	NOUN
ejpam-4396	8	29	will	will	AUX
ejpam-4396	8	30	be	be	AUX
ejpam-4396	8	31	denoted	denote	VERB
ejpam-4396	8	32	the	the	PRON
ejpam-4396	8	33	by	by	ADP
ejpam-4396	8	34	a	a	DET
ejpam-4396	8	35	∗	∗	X
ejpam-4396	8	36	b.	b.	NOUN
ejpam-4396	9	1	a	a	DET
ejpam-4396	9	2	nonempty	nonempty	ADV
ejpam-4396	9	3	set	set	VERB
ejpam-4396	9	4	g	g	NOUN
ejpam-4396	9	5	is	be	AUX
ejpam-4396	9	6	a	a	DET
ejpam-4396	9	7	g	g	NOUN
ejpam-4396	9	8	-	-	PUNCT
ejpam-4396	9	9	group	group	NOUN
ejpam-4396	9	10	with	with	ADP
ejpam-4396	9	11	respect	respect	NOUN
ejpam-4396	9	12	to	to	ADP
ejpam-4396	9	13	a	a	DET
ejpam-4396	9	14	binary	binary	ADJ
ejpam-4396	9	15	operation	operation	NOUN
ejpam-4396	9	16	∗	∗	NOUN
ejpam-4396	9	17	if	if	SCONJ
ejpam-4396	9	18	it	it	PRON
ejpam-4396	9	19	satisfies	satisfy	VERB
ejpam-4396	9	20	the	the	DET
ejpam-4396	9	21	following	follow	VERB
ejpam-4396	9	22	properties	property	NOUN
ejpam-4396	9	23	:	:	PUNCT
ejpam-4396	9	24	(	(	PUNCT
ejpam-4396	9	25	g1	g1	PROPN
ejpam-4396	9	26	)	)	PUNCT
ejpam-4396	9	27	a	a	DET
ejpam-4396	9	28	∗	∗	NOUN
ejpam-4396	9	29	(	(	PUNCT
ejpam-4396	9	30	b	b	NOUN
ejpam-4396	9	31	∗	∗	NOUN
ejpam-4396	9	32	c	c	NOUN
ejpam-4396	9	33	)	)	PUNCT
ejpam-4396	10	1	=	=	NOUN
ejpam-4396	10	2	(	(	PUNCT
ejpam-4396	10	3	a	a	DET
ejpam-4396	10	4	∗	∗	NOUN
ejpam-4396	10	5	b	b	NOUN
ejpam-4396	10	6	)	)	PUNCT
ejpam-4396	10	7	∗	∗	NOUN
ejpam-4396	10	8	c	c	NOUN
ejpam-4396	10	9	for	for	ADP
ejpam-4396	10	10	all	all	DET
ejpam-4396	10	11	a	a	DET
ejpam-4396	10	12	,	,	PUNCT
ejpam-4396	10	13	b	b	NOUN
ejpam-4396	10	14	,	,	PUNCT
ejpam-4396	10	15	c	c	PROPN
ejpam-4396	10	16	∈	∈	PROPN
ejpam-4396	10	17	g	g	PROPN
ejpam-4396	10	18	(	(	PUNCT
ejpam-4396	10	19	in	in	ADP
ejpam-4396	10	20	this	this	DET
ejpam-4396	10	21	case	case	NOUN
ejpam-4396	10	22	,	,	PUNCT
ejpam-4396	10	23	we	we	PRON
ejpam-4396	10	24	say	say	VERB
ejpam-4396	10	25	that	that	SCONJ
ejpam-4396	10	26	∗	∗	NOUN
ejpam-4396	10	27	is	be	AUX
ejpam-4396	10	28	associative	associative	ADJ
ejpam-4396	10	29	)	)	PUNCT
ejpam-4396	10	30	;	;	PUNCT
ejpam-4396	10	31	(	(	PUNCT
ejpam-4396	10	32	g2	g2	PROPN
ejpam-4396	10	33	)	)	PUNCT
ejpam-4396	10	34	for	for	ADP
ejpam-4396	10	35	each	each	PRON
ejpam-4396	10	36	a	a	DET
ejpam-4396	10	37	∈	∈	PROPN
ejpam-4396	10	38	g	g	NOUN
ejpam-4396	10	39	,	,	PUNCT
ejpam-4396	10	40	there	there	PRON
ejpam-4396	10	41	exists	exist	VERB
ejpam-4396	10	42	an	an	DET
ejpam-4396	10	43	element	element	NOUN
ejpam-4396	10	44	e	e	PROPN
ejpam-4396	10	45	∈	∈	PROPN
ejpam-4396	10	46	g	g	PROPN
ejpam-4396	10	47	(	(	PUNCT
ejpam-4396	10	48	called	call	VERB
ejpam-4396	10	49	an	an	DET
ejpam-4396	10	50	identity	identity	NOUN
ejpam-4396	10	51	element	element	NOUN
ejpam-4396	10	52	)	)	PUNCT
ejpam-4396	10	53	such	such	ADJ
ejpam-4396	10	54	that	that	SCONJ
ejpam-4396	10	55	a	a	DET
ejpam-4396	10	56	∗	∗	NOUN
ejpam-4396	10	57	e	e	NOUN
ejpam-4396	10	58	=	=	NOUN
ejpam-4396	10	59	a	a	DET
ejpam-4396	10	60	=	=	SYM
ejpam-4396	10	61	e	e	PROPN
ejpam-4396	10	62	∗	∗	NOUN
ejpam-4396	10	63	a	a	PRON
ejpam-4396	10	64	;	;	PUNCT
ejpam-4396	10	65	and	and	CCONJ
ejpam-4396	10	66	,	,	PUNCT
ejpam-4396	10	67	(	(	PUNCT
ejpam-4396	10	68	g3	g3	NOUN
ejpam-4396	10	69	)	)	PUNCT
ejpam-4396	10	70	for	for	ADP
ejpam-4396	10	71	each	each	PRON
ejpam-4396	10	72	a	a	DET
ejpam-4396	10	73	∈	∈	PROPN
ejpam-4396	10	74	g	g	NOUN
ejpam-4396	10	75	,	,	PUNCT
ejpam-4396	10	76	there	there	PRON
ejpam-4396	10	77	exists	exist	VERB
ejpam-4396	10	78	an	an	DET
ejpam-4396	10	79	element	element	NOUN
ejpam-4396	10	80	b	b	PROPN
ejpam-4396	10	81	∈	∈	PROPN
ejpam-4396	10	82	g	g	PROPN
ejpam-4396	10	83	(	(	PUNCT
ejpam-4396	10	84	called	call	VERB
ejpam-4396	10	85	an	an	DET
ejpam-4396	10	86	inverse	inverse	NOUN
ejpam-4396	10	87	of	of	ADP
ejpam-4396	10	88	g	g	NOUN
ejpam-4396	10	89	)	)	PUNCT
ejpam-4396	10	90	such	such	ADJ
ejpam-4396	10	91	that	that	SCONJ
ejpam-4396	10	92	a	a	DET
ejpam-4396	10	93	∗	∗	NOUN
ejpam-4396	10	94	b	b	NOUN
ejpam-4396	10	95	=	=	SYM
ejpam-4396	10	96	e	e	NOUN
ejpam-4396	10	97	=	=	SYM
ejpam-4396	10	98	b	b	PROPN
ejpam-4396	10	99	∗a	∗a	ADJ
ejpam-4396	10	100	for	for	ADP
ejpam-4396	10	101	some	some	DET
ejpam-4396	10	102	identity	identity	NOUN
ejpam-4396	10	103	element	element	NOUN
ejpam-4396	10	104	e	e	PROPN
ejpam-4396	10	105	of	of	ADP
ejpam-4396	10	106	a.	a.	NOUN
ejpam-4396	10	107	in	in	ADP
ejpam-4396	10	108	this	this	DET
ejpam-4396	10	109	case	case	NOUN
ejpam-4396	10	110	,	,	PUNCT
ejpam-4396	10	111	we	we	PRON
ejpam-4396	10	112	write	write	VERB
ejpam-4396	10	113	(	(	PUNCT
ejpam-4396	10	114	g	g	NOUN
ejpam-4396	10	115	,	,	PUNCT
ejpam-4396	10	116	∗	∗	NOUN
ejpam-4396	10	117	)	)	PUNCT
ejpam-4396	10	118	to	to	PART
ejpam-4396	10	119	denote	denote	VERB
ejpam-4396	10	120	the	the	DET
ejpam-4396	10	121	algebraic	algebraic	ADJ
ejpam-4396	10	122	structure	structure	NOUN
ejpam-4396	10	123	.	.	PUNCT
ejpam-4396	11	1	if	if	SCONJ
ejpam-4396	11	2	a	a	DET
ejpam-4396	11	3	∗	∗	X
ejpam-4396	11	4	b	b	NOUN
ejpam-4396	11	5	=	=	SYM
ejpam-4396	11	6	b	b	PROPN
ejpam-4396	11	7	∗	∗	NOUN
ejpam-4396	11	8	a	a	PRON
ejpam-4396	11	9	for	for	ADP
ejpam-4396	11	10	all	all	DET
ejpam-4396	11	11	a	a	PRON
ejpam-4396	11	12	,	,	PUNCT
ejpam-4396	11	13	b	b	X
ejpam-4396	11	14	∈	∈	PROPN
ejpam-4396	11	15	g	g	PROPN
ejpam-4396	11	16	,	,	PUNCT
ejpam-4396	11	17	then	then	ADV
ejpam-4396	11	18	we	we	PRON
ejpam-4396	11	19	say	say	VERB
ejpam-4396	11	20	that	that	SCONJ
ejpam-4396	11	21	g	g	PROPN
ejpam-4396	11	22	is	be	AUX
ejpam-4396	11	23	an	an	DET
ejpam-4396	11	24	abelian	abelian	ADJ
ejpam-4396	11	25	g	g	NOUN
ejpam-4396	11	26	-	-	PUNCT
ejpam-4396	11	27	group	group	NOUN
ejpam-4396	11	28	.	.	PUNCT
ejpam-4396	12	1	an	an	DET
ejpam-4396	12	2	element	element	NOUN
ejpam-4396	12	3	with	with	ADP
ejpam-4396	12	4	a	a	DET
ejpam-4396	12	5	unique	unique	ADJ
ejpam-4396	12	6	identity	identity	NOUN
ejpam-4396	12	7	element	element	NOUN
ejpam-4396	12	8	is	be	AUX
ejpam-4396	12	9	called	call	VERB
ejpam-4396	12	10	a	a	DET
ejpam-4396	12	11	unit	unit	NOUN
ejpam-4396	12	12	,	,	PUNCT
ejpam-4396	12	13	otherwise	otherwise	ADV
ejpam-4396	12	14	we	we	PRON
ejpam-4396	12	15	say	say	VERB
ejpam-4396	12	16	that	that	SCONJ
ejpam-4396	12	17	a	a	PRON
ejpam-4396	12	18	is	be	AUX
ejpam-4396	12	19	non	non	ADJ
ejpam-4396	12	20	-	-	NOUN
ejpam-4396	12	21	unit	unit	NOUN
ejpam-4396	12	22	.	.	PUNCT
ejpam-4396	13	1	the	the	DET
ejpam-4396	13	2	singleton	singleton	PROPN
ejpam-4396	13	3	sets	set	VERB
ejpam-4396	13	4	{	{	PUNCT
ejpam-4396	13	5	0	0	NUM
ejpam-4396	13	6	}	}	PUNCT
ejpam-4396	13	7	and	and	CCONJ
ejpam-4396	13	8	{	{	PUNCT
ejpam-4396	13	9	1	1	X
ejpam-4396	13	10	}	}	PUNCT
ejpam-4396	13	11	with	with	ADP
ejpam-4396	13	12	respect	respect	NOUN
ejpam-4396	13	13	to	to	ADP
ejpam-4396	13	14	multiplication	multiplication	NOUN
ejpam-4396	13	15	×	×	NOUN
ejpam-4396	13	16	are	be	AUX
ejpam-4396	13	17	g	g	NOUN
ejpam-4396	13	18	-	-	PUNCT
ejpam-4396	13	19	groups	group	NOUN
ejpam-4396	13	20	(	(	PUNCT
ejpam-4396	13	21	the	the	DET
ejpam-4396	13	22	two	two	NUM
ejpam-4396	13	23	are	be	AUX
ejpam-4396	13	24	called	call	VERB
ejpam-4396	13	25	trivial	trivial	ADJ
ejpam-4396	13	26	g	g	NOUN
ejpam-4396	13	27	-	-	PUNCT
ejpam-4396	13	28	groups	group	NOUN
ejpam-4396	13	29	)	)	PUNCT
ejpam-4396	13	30	.	.	PUNCT
ejpam-4396	14	1	tables	table	NOUN
ejpam-4396	14	2	1	1	NUM
ejpam-4396	14	3	and	and	CCONJ
ejpam-4396	14	4	2	2	NUM
ejpam-4396	14	5	may	may	AUX
ejpam-4396	14	6	be	be	AUX
ejpam-4396	14	7	helpful	helpful	ADJ
ejpam-4396	14	8	in	in	ADP
ejpam-4396	14	9	seeing	see	VERB
ejpam-4396	14	10	this	this	PRON
ejpam-4396	14	11	.	.	PUNCT
ejpam-4396	15	1	also	also	ADV
ejpam-4396	15	2	,	,	PUNCT
ejpam-4396	15	3	the	the	DET
ejpam-4396	15	4	set	set	NOUN
ejpam-4396	15	5	{	{	PUNCT
ejpam-4396	15	6	0	0	NUM
ejpam-4396	15	7	,	,	PUNCT
ejpam-4396	15	8	1	1	NUM
ejpam-4396	15	9	}	}	PUNCT
ejpam-4396	15	10	is	be	AUX
ejpam-4396	15	11	also	also	ADV
ejpam-4396	15	12	a	a	DET
ejpam-4396	15	13	g	g	NOUN
ejpam-4396	15	14	-	-	PUNCT
ejpam-4396	15	15	group	group	NOUN
ejpam-4396	15	16	under	under	ADP
ejpam-4396	15	17	multiplication	multiplication	NOUN
ejpam-4396	15	18	as	as	SCONJ
ejpam-4396	15	19	shown	show	VERB
ejpam-4396	15	20	in	in	ADP
ejpam-4396	15	21	table	table	NOUN
ejpam-4396	15	22	3	3	NUM
ejpam-4396	15	23	.	.	PUNCT
ejpam-4396	16	1	the	the	DET
ejpam-4396	16	2	introduction	introduction	NOUN
ejpam-4396	16	3	of	of	ADP
ejpam-4396	16	4	the	the	DET
ejpam-4396	16	5	algebraic	algebraic	ADJ
ejpam-4396	16	6	structure	structure	NOUN
ejpam-4396	16	7	g	g	PROPN
ejpam-4396	16	8	-	-	PUNCT
ejpam-4396	16	9	group	group	NOUN
ejpam-4396	16	10	was	be	AUX
ejpam-4396	16	11	motivated	motivate	VERB
ejpam-4396	16	12	by	by	ADP
ejpam-4396	16	13	the	the	DET
ejpam-4396	16	14	intention	intention	NOUN
ejpam-4396	16	15	of	of	ADP
ejpam-4396	16	16	presenting	present	VERB
ejpam-4396	16	17	a	a	DET
ejpam-4396	16	18	structure	structure	NOUN
ejpam-4396	16	19	having	have	VERB
ejpam-4396	16	20	a	a	DET
ejpam-4396	16	21	unique	unique	ADJ
ejpam-4396	16	22	operation	operation	NOUN
ejpam-4396	16	23	which	which	PRON
ejpam-4396	16	24	generalizes	generalize	VERB
ejpam-4396	16	25	the	the	DET
ejpam-4396	16	26	properties	property	NOUN
ejpam-4396	16	27	of	of	ADP
ejpam-4396	16	28	the	the	DET
ejpam-4396	16	29	operation	operation	NOUN
ejpam-4396	16	30	multiplication	multiplication	NOUN
ejpam-4396	16	31	in	in	ADP
ejpam-4396	16	32	a	a	DET
ejpam-4396	16	33	field	field	NOUN
ejpam-4396	16	34	.	.	PUNCT
ejpam-4396	17	1	∗corresponding	∗corresponde	VERB
ejpam-4396	17	2	author	author	NOUN
ejpam-4396	17	3	.	.	PUNCT
ejpam-4396	18	1	doi	doi	NOUN
ejpam-4396	18	2	:	:	PUNCT
ejpam-4396	18	3	https://doi.org/10.29020/nybg.ejpam.v15i3.4396	https://doi.org/10.29020/nybg.ejpam.v15i3.4396	PRON
ejpam-4396	18	4	email	email	NOUN
ejpam-4396	18	5	addresses	address	NOUN
ejpam-4396	18	6	:	:	PUNCT
ejpam-4396	18	7	jcaraquil@southernleytestateu.edu.ph	jcaraquil@southernleytestateu.edu.ph	PROPN
ejpam-4396	18	8	(	(	PUNCT
ejpam-4396	18	9	j.	j.	PROPN
ejpam-4396	18	10	caraquil	caraquil	PROPN
ejpam-4396	18	11	)	)	PUNCT
ejpam-4396	18	12	,	,	PUNCT
ejpam-4396	18	13	michael.baldadojr@norsu.edu.ph	michael.baldadojr@norsu.edu.ph	PROPN
ejpam-4396	18	14	(	(	PUNCT
ejpam-4396	18	15	m.	m.	PROPN
ejpam-4396	18	16	baldado	baldado	PROPN
ejpam-4396	18	17	jr	jr	PROPN
ejpam-4396	18	18	.	.	PUNCT
ejpam-4396	18	19	)	)	PUNCT
ejpam-4396	18	20	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4396	19	1	887	887	NUM
ejpam-4396	19	2	©	©	PROPN
ejpam-4396	19	3	2022	2022	NUM
ejpam-4396	19	4	ejpam	ejpam	VERB
ejpam-4396	19	5	all	all	DET
ejpam-4396	19	6	rights	right	NOUN
ejpam-4396	19	7	reserved	reserve	VERB
ejpam-4396	19	8	.	.	PUNCT
ejpam-4396	20	1	j.	j.	PROPN
ejpam-4396	20	2	caraquil	caraquil	PROPN
ejpam-4396	20	3	,	,	PUNCT
ejpam-4396	20	4	m.	m.	PROPN
ejpam-4396	20	5	baldado	baldado	PROPN
ejpam-4396	20	6	jr	jr	PROPN
ejpam-4396	20	7	.	.	PROPN
ejpam-4396	20	8	/	/	SYM
ejpam-4396	20	9	eur	eur	PROPN
ejpam-4396	20	10	.	.	PUNCT
ejpam-4396	21	1	j.	j.	PROPN
ejpam-4396	21	2	pure	pure	PROPN
ejpam-4396	21	3	appl	appl	PROPN
ejpam-4396	21	4	.	.	PROPN
ejpam-4396	21	5	math	math	PROPN
ejpam-4396	21	6	,	,	PUNCT
ejpam-4396	21	7	15	15	NUM
ejpam-4396	21	8	(	(	PUNCT
ejpam-4396	21	9	3	3	NUM
ejpam-4396	21	10	)	)	PUNCT
ejpam-4396	21	11	(	(	PUNCT
ejpam-4396	21	12	2022	2022	NUM
ejpam-4396	21	13	)	)	PUNCT
ejpam-4396	21	14	,	,	PUNCT
ejpam-4396	21	15	887	887	NUM
ejpam-4396	21	16	-	-	SYM
ejpam-4396	21	17	896	896	NUM
ejpam-4396	21	18	888	888	NUM
ejpam-4396	21	19	×	×	NOUN
ejpam-4396	21	20	0	0	NUM
ejpam-4396	21	21	0	0	NUM
ejpam-4396	21	22	0	0	NUM
ejpam-4396	21	23	table	table	NOUN
ejpam-4396	21	24	1	1	NUM
ejpam-4396	21	25	:	:	PUNCT
ejpam-4396	21	26	the	the	DET
ejpam-4396	21	27	g	g	PROPN
ejpam-4396	21	28	-	-	PUNCT
ejpam-4396	21	29	group	group	NOUN
ejpam-4396	21	30	{	{	PUNCT
ejpam-4396	21	31	0	0	NUM
ejpam-4396	21	32	}	}	PUNCT
ejpam-4396	21	33	×	×	NOUN
ejpam-4396	21	34	1	1	NUM
ejpam-4396	21	35	1	1	NUM
ejpam-4396	21	36	1	1	NUM
ejpam-4396	21	37	table	table	NOUN
ejpam-4396	21	38	2	2	NUM
ejpam-4396	21	39	:	:	PUNCT
ejpam-4396	21	40	the	the	DET
ejpam-4396	21	41	g	g	PROPN
ejpam-4396	21	42	-	-	PUNCT
ejpam-4396	21	43	group	group	NOUN
ejpam-4396	21	44	{	{	PUNCT
ejpam-4396	21	45	1	1	NUM
ejpam-4396	21	46	}	}	PUNCT
ejpam-4396	21	47	let	let	VERB
ejpam-4396	21	48	g	g	NOUN
ejpam-4396	21	49	be	be	AUX
ejpam-4396	21	50	a	a	DET
ejpam-4396	21	51	non	non	ADJ
ejpam-4396	21	52	-	-	ADJ
ejpam-4396	21	53	empty	empty	ADJ
ejpam-4396	21	54	set	set	NOUN
ejpam-4396	21	55	.	.	PUNCT
ejpam-4396	22	1	an	an	DET
ejpam-4396	22	2	e	e	NOUN
ejpam-4396	22	3	-	-	NOUN
ejpam-4396	22	4	group	group	NOUN
ejpam-4396	22	5	is	be	AUX
ejpam-4396	22	6	an	an	DET
ejpam-4396	22	7	algebra	algebra	NOUN
ejpam-4396	22	8	(	(	PUNCT
ejpam-4396	22	9	g	g	NOUN
ejpam-4396	22	10	;	;	PUNCT
ejpam-4396	22	11	∗;a	∗;a	NOUN
ejpam-4396	22	12	)	)	PUNCT
ejpam-4396	22	13	where	where	SCONJ
ejpam-4396	22	14	∗	∗	NOUN
ejpam-4396	22	15	is	be	AUX
ejpam-4396	22	16	a	a	DET
ejpam-4396	22	17	binary	binary	ADJ
ejpam-4396	22	18	operation	operation	NOUN
ejpam-4396	22	19	in	in	ADP
ejpam-4396	22	20	g	g	PROPN
ejpam-4396	22	21	and	and	CCONJ
ejpam-4396	22	22	a	a	PRON
ejpam-4396	22	23	is	be	AUX
ejpam-4396	22	24	a	a	DET
ejpam-4396	22	25	non	non	ADJ
ejpam-4396	22	26	-	-	ADJ
ejpam-4396	22	27	empty	empty	ADJ
ejpam-4396	22	28	subset	subset	NOUN
ejpam-4396	22	29	of	of	ADP
ejpam-4396	22	30	g	g	NOUN
ejpam-4396	22	31	which	which	PRON
ejpam-4396	22	32	satisfies	satisfy	VERB
ejpam-4396	22	33	the	the	DET
ejpam-4396	22	34	following	follow	VERB
ejpam-4396	22	35	axioms	axiom	NOUN
ejpam-4396	22	36	:	:	PUNCT
ejpam-4396	22	37	(	(	PUNCT
ejpam-4396	22	38	e1	e1	PROPN
ejpam-4396	22	39	)	)	PUNCT
ejpam-4396	22	40	x∗(y∗z	x∗(y∗z	NUM
ejpam-4396	22	41	)	)	PUNCT
ejpam-4396	23	1	=	=	PRON
ejpam-4396	23	2	(	(	PUNCT
ejpam-4396	23	3	x∗y)∗z	x∗y)∗z	NOUN
ejpam-4396	23	4	for	for	ADP
ejpam-4396	23	5	all	all	DET
ejpam-4396	23	6	x	x	PROPN
ejpam-4396	23	7	,	,	PUNCT
ejpam-4396	23	8	y	y	PROPN
ejpam-4396	23	9	,	,	PUNCT
ejpam-4396	23	10	z	z	PROPN
ejpam-4396	23	11	∈	∈	PROPN
ejpam-4396	23	12	g	g	NOUN
ejpam-4396	23	13	;	;	PUNCT
ejpam-4396	23	14	(	(	PUNCT
ejpam-4396	23	15	e2	e2	PROPN
ejpam-4396	23	16	)	)	PUNCT
ejpam-4396	23	17	for	for	ADP
ejpam-4396	23	18	every	every	DET
ejpam-4396	23	19	x	x	SYM
ejpam-4396	23	20	∈	∈	PROPN
ejpam-4396	23	21	g	g	NOUN
ejpam-4396	23	22	there	there	PRON
ejpam-4396	23	23	exists	exist	VERB
ejpam-4396	23	24	an	an	DET
ejpam-4396	23	25	element	element	NOUN
ejpam-4396	23	26	a	a	DET
ejpam-4396	23	27	∈	∈	PROPN
ejpam-4396	23	28	a	a	DET
ejpam-4396	23	29	such	such	ADJ
ejpam-4396	23	30	that	that	SCONJ
ejpam-4396	23	31	x	x	PUNCT
ejpam-4396	23	32	∗	∗	VERB
ejpam-4396	23	33	a	a	PRON
ejpam-4396	23	34	=	=	NOUN
ejpam-4396	23	35	a	a	DET
ejpam-4396	23	36	∗	∗	NOUN
ejpam-4396	23	37	x	x	X
ejpam-4396	23	38	=	=	SYM
ejpam-4396	23	39	x	x	X
ejpam-4396	23	40	(	(	PUNCT
ejpam-4396	23	41	the	the	DET
ejpam-4396	23	42	existence	existence	NOUN
ejpam-4396	23	43	of	of	ADP
ejpam-4396	23	44	an	an	DET
ejpam-4396	23	45	identity	identity	NOUN
ejpam-4396	23	46	element	element	NOUN
ejpam-4396	23	47	corresponding	correspond	VERB
ejpam-4396	23	48	to	to	ADP
ejpam-4396	23	49	every	every	DET
ejpam-4396	23	50	element	element	NOUN
ejpam-4396	23	51	of	of	ADP
ejpam-4396	23	52	g	g	PROPN
ejpam-4396	23	53	)	)	PUNCT
ejpam-4396	23	54	;	;	PUNCT
ejpam-4396	23	55	and	and	CCONJ
ejpam-4396	23	56	,	,	PUNCT
ejpam-4396	23	57	(	(	PUNCT
ejpam-4396	23	58	e3	e3	NOUN
ejpam-4396	23	59	)	)	PUNCT
ejpam-4396	23	60	for	for	ADP
ejpam-4396	23	61	every	every	DET
ejpam-4396	23	62	x	x	SYM
ejpam-4396	23	63	∈	∈	PROPN
ejpam-4396	23	64	g	g	NOUN
ejpam-4396	23	65	there	there	PRON
ejpam-4396	23	66	exists	exist	VERB
ejpam-4396	23	67	an	an	DET
ejpam-4396	23	68	element	element	NOUN
ejpam-4396	23	69	y	y	PROPN
ejpam-4396	23	70	∈	∈	PROPN
ejpam-4396	23	71	g	g	PROPN
ejpam-4396	23	72	such	such	ADJ
ejpam-4396	23	73	that	that	SCONJ
ejpam-4396	23	74	x	x	PROPN
ejpam-4396	23	75	∗	∗	PROPN
ejpam-4396	23	76	y	y	PROPN
ejpam-4396	23	77	,	,	PUNCT
ejpam-4396	23	78	y	y	PROPN
ejpam-4396	23	79	∗	∗	NOUN
ejpam-4396	23	80	x	x	PUNCT
ejpam-4396	23	81	∈	∈	PROPN
ejpam-4396	23	82	a	a	PRON
ejpam-4396	24	1	[	[	X
ejpam-4396	24	2	6	6	NUM
ejpam-4396	24	3	]	]	PUNCT
ejpam-4396	24	4	.	.	PUNCT
ejpam-4396	25	1	let	let	VERB
ejpam-4396	25	2	u	u	PRON
ejpam-4396	25	3	be	be	AUX
ejpam-4396	25	4	a	a	DET
ejpam-4396	25	5	non	non	ADJ
ejpam-4396	25	6	-	-	ADJ
ejpam-4396	25	7	empty	empty	ADJ
ejpam-4396	25	8	set	set	NOUN
ejpam-4396	25	9	,	,	PUNCT
ejpam-4396	25	10	and	and	CCONJ
ejpam-4396	25	11	∗	∗	NOUN
ejpam-4396	25	12	be	be	VERB
ejpam-4396	25	13	a	a	DET
ejpam-4396	25	14	binary	binary	ADJ
ejpam-4396	25	15	operation	operation	NOUN
ejpam-4396	25	16	in	in	ADP
ejpam-4396	25	17	u	u	PROPN
ejpam-4396	25	18	.	.	PUNCT
ejpam-4396	26	1	the	the	DET
ejpam-4396	26	2	couple	couple	PROPN
ejpam-4396	26	3	⟨u	⟨u	NOUN
ejpam-4396	26	4	,	,	PUNCT
ejpam-4396	26	5	∗⟩	∗⟩	PRON
ejpam-4396	26	6	is	be	AUX
ejpam-4396	26	7	an	an	DET
ejpam-4396	26	8	ubat	ubat	ADJ
ejpam-4396	26	9	-	-	PUNCT
ejpam-4396	26	10	space	space	NOUN
ejpam-4396	26	11	if	if	SCONJ
ejpam-4396	26	12	the	the	DET
ejpam-4396	26	13	following	follow	VERB
ejpam-4396	26	14	properties	property	NOUN
ejpam-4396	26	15	hold	hold	VERB
ejpam-4396	26	16	:	:	PUNCT
ejpam-4396	26	17	(	(	PUNCT
ejpam-4396	26	18	u1	u1	NOUN
ejpam-4396	26	19	)	)	PUNCT
ejpam-4396	27	1	x	x	SYM
ejpam-4396	27	2	∗	∗	NOUN
ejpam-4396	27	3	(	(	PUNCT
ejpam-4396	27	4	y	y	PROPN
ejpam-4396	27	5	∗	∗	PROPN
ejpam-4396	27	6	z	z	NOUN
ejpam-4396	27	7	)	)	PUNCT
ejpam-4396	27	8	=	=	SYM
ejpam-4396	27	9	(	(	PUNCT
ejpam-4396	27	10	x	x	X
ejpam-4396	27	11	∗	∗	PROPN
ejpam-4396	27	12	y	y	NOUN
ejpam-4396	27	13	)	)	PUNCT
ejpam-4396	27	14	∗	∗	NOUN
ejpam-4396	27	15	z	z	NOUN
ejpam-4396	27	16	for	for	ADP
ejpam-4396	27	17	all	all	DET
ejpam-4396	27	18	x	x	NOUN
ejpam-4396	27	19	,	,	PUNCT
ejpam-4396	27	20	y	y	PROPN
ejpam-4396	27	21	,	,	PUNCT
ejpam-4396	27	22	z	z	PROPN
ejpam-4396	27	23	∈	∈	PROPN
ejpam-4396	27	24	u	u	NOUN
ejpam-4396	27	25	;	;	PUNCT
ejpam-4396	27	26	(	(	PUNCT
ejpam-4396	27	27	u2	u2	NOUN
ejpam-4396	27	28	)	)	PUNCT
ejpam-4396	27	29	there	there	PRON
ejpam-4396	27	30	exists	exist	VERB
ejpam-4396	27	31	y	y	PROPN
ejpam-4396	27	32	∈	∈	PROPN
ejpam-4396	27	33	u	u	NOUN
ejpam-4396	27	34	such	such	ADJ
ejpam-4396	27	35	that	that	SCONJ
ejpam-4396	27	36	x	x	PROPN
ejpam-4396	27	37	∗	∗	NOUN
ejpam-4396	27	38	y	y	NOUN
ejpam-4396	27	39	=	=	SYM
ejpam-4396	27	40	y	y	PROPN
ejpam-4396	27	41	∗	∗	NOUN
ejpam-4396	27	42	x	x	PUNCT
ejpam-4396	27	43	=	=	SYM
ejpam-4396	27	44	y	y	PROPN
ejpam-4396	27	45	for	for	ADP
ejpam-4396	27	46	all	all	DET
ejpam-4396	27	47	x	x	SYM
ejpam-4396	27	48	∈	∈	PROPN
ejpam-4396	27	49	u	u	NOUN
ejpam-4396	27	50	;	;	PUNCT
ejpam-4396	27	51	and	and	CCONJ
ejpam-4396	27	52	,	,	PUNCT
ejpam-4396	27	53	(	(	PUNCT
ejpam-4396	27	54	u3	u3	NOUN
ejpam-4396	27	55	)	)	PUNCT
ejpam-4396	27	56	there	there	PRON
ejpam-4396	27	57	exists	exist	VERB
ejpam-4396	27	58	z	z	PROPN
ejpam-4396	27	59	∈	∈	PROPN
ejpam-4396	27	60	u	u	NOUN
ejpam-4396	27	61	such	such	ADJ
ejpam-4396	27	62	that	that	SCONJ
ejpam-4396	27	63	x	x	PROPN
ejpam-4396	27	64	∗	∗	NOUN
ejpam-4396	27	65	z	z	NOUN
ejpam-4396	28	1	=	=	SYM
ejpam-4396	28	2	z	z	NOUN
ejpam-4396	28	3	∗	∗	NOUN
ejpam-4396	28	4	x	x	X
ejpam-4396	29	1	=	=	PUNCT
ejpam-4396	29	2	x	x	PROPN
ejpam-4396	29	3	for	for	ADP
ejpam-4396	29	4	all	all	DET
ejpam-4396	29	5	x	x	SYM
ejpam-4396	29	6	∈	∈	PROPN
ejpam-4396	29	7	u	u	NOUN
ejpam-4396	29	8	.	.	PUNCT
ejpam-4396	30	1	for	for	ADP
ejpam-4396	30	2	example	example	NOUN
ejpam-4396	30	3	,	,	PUNCT
ejpam-4396	30	4	the	the	DET
ejpam-4396	30	5	singleton	singleton	PROPN
ejpam-4396	30	6	set	set	NOUN
ejpam-4396	30	7	{	{	PUNCT
ejpam-4396	30	8	0	0	NUM
ejpam-4396	30	9	}	}	PUNCT
ejpam-4396	30	10	with	with	ADP
ejpam-4396	30	11	respect	respect	NOUN
ejpam-4396	30	12	to	to	ADP
ejpam-4396	30	13	multiplication	multiplication	VERB
ejpam-4396	30	14	×	×	NOUN
ejpam-4396	30	15	in	in	ADP
ejpam-4396	30	16	the	the	DET
ejpam-4396	30	17	table	table	NOUN
ejpam-4396	30	18	1	1	NUM
ejpam-4396	30	19	,	,	PUNCT
ejpam-4396	30	20	the	the	DET
ejpam-4396	30	21	singleton	singleton	NOUN
ejpam-4396	30	22	set	set	NOUN
ejpam-4396	30	23	{	{	PUNCT
ejpam-4396	30	24	1	1	NUM
ejpam-4396	30	25	}	}	PUNCT
ejpam-4396	30	26	with	with	ADP
ejpam-4396	30	27	respect	respect	NOUN
ejpam-4396	30	28	to	to	ADP
ejpam-4396	30	29	multiplication	multiplication	VERB
ejpam-4396	30	30	×	×	NOUN
ejpam-4396	30	31	in	in	ADP
ejpam-4396	30	32	the	the	DET
ejpam-4396	30	33	table	table	NOUN
ejpam-4396	30	34	2	2	NUM
ejpam-4396	30	35	and	and	CCONJ
ejpam-4396	30	36	the	the	DET
ejpam-4396	30	37	set	set	NOUN
ejpam-4396	30	38	{	{	PUNCT
ejpam-4396	30	39	0	0	NUM
ejpam-4396	30	40	,	,	PUNCT
ejpam-4396	30	41	1	1	NUM
ejpam-4396	30	42	}	}	PUNCT
ejpam-4396	30	43	under	under	ADP
ejpam-4396	30	44	multiplication	multiplication	NOUN
ejpam-4396	30	45	in	in	ADP
ejpam-4396	30	46	table	table	NOUN
ejpam-4396	30	47	3	3	NUM
ejpam-4396	30	48	are	be	AUX
ejpam-4396	30	49	also	also	ADV
ejpam-4396	30	50	ubat	ubat	ADJ
ejpam-4396	30	51	-	-	PUNCT
ejpam-4396	30	52	spaces	space	NOUN
ejpam-4396	30	53	.	.	PUNCT
ejpam-4396	31	1	a	a	DET
ejpam-4396	31	2	nonempty	nonempty	ADV
ejpam-4396	31	3	set	set	VERB
ejpam-4396	31	4	g	g	NOUN
ejpam-4396	31	5	is	be	AUX
ejpam-4396	31	6	a	a	DET
ejpam-4396	31	7	generalized	generalized	ADJ
ejpam-4396	31	8	group	group	NOUN
ejpam-4396	31	9	with	with	ADP
ejpam-4396	31	10	respect	respect	NOUN
ejpam-4396	31	11	to	to	ADP
ejpam-4396	31	12	a	a	DET
ejpam-4396	31	13	binary	binary	ADJ
ejpam-4396	31	14	operation	operation	NOUN
ejpam-4396	31	15	∗	∗	NOUN
ejpam-4396	31	16	if	if	SCONJ
ejpam-4396	31	17	it	it	PRON
ejpam-4396	31	18	satisfies	satisfy	VERB
ejpam-4396	31	19	the	the	DET
ejpam-4396	31	20	following	follow	VERB
ejpam-4396	31	21	properties	property	NOUN
ejpam-4396	31	22	.	.	PUNCT
ejpam-4396	32	1	(	(	PUNCT
ejpam-4396	32	2	m1	m1	NOUN
ejpam-4396	32	3	)	)	PUNCT
ejpam-4396	32	4	f	f	PROPN
ejpam-4396	32	5	∗	∗	NOUN
ejpam-4396	32	6	(	(	PUNCT
ejpam-4396	32	7	g	g	PROPN
ejpam-4396	32	8	∗	∗	NOUN
ejpam-4396	32	9	h	h	NOUN
ejpam-4396	32	10	)	)	PUNCT
ejpam-4396	32	11	=	=	PUNCT
ejpam-4396	32	12	(	(	PUNCT
ejpam-4396	32	13	f	f	PROPN
ejpam-4396	32	14	∗	∗	X
ejpam-4396	32	15	g	g	NOUN
ejpam-4396	32	16	)	)	PUNCT
ejpam-4396	32	17	∗	∗	NOUN
ejpam-4396	32	18	h	h	NOUN
ejpam-4396	32	19	for	for	ADP
ejpam-4396	32	20	all	all	DET
ejpam-4396	32	21	f	f	PROPN
ejpam-4396	32	22	,	,	PUNCT
ejpam-4396	32	23	g	g	PROPN
ejpam-4396	32	24	,	,	PUNCT
ejpam-4396	32	25	h	h	NOUN
ejpam-4396	32	26	∈	∈	PROPN
ejpam-4396	32	27	g	g	NOUN
ejpam-4396	32	28	;	;	PUNCT
ejpam-4396	32	29	(	(	PUNCT
ejpam-4396	32	30	m2	m2	PROPN
ejpam-4396	32	31	)	)	PUNCT
ejpam-4396	32	32	for	for	ADP
ejpam-4396	32	33	each	each	DET
ejpam-4396	32	34	g	g	PROPN
ejpam-4396	32	35	∈	∈	PROPN
ejpam-4396	32	36	g	g	NOUN
ejpam-4396	32	37	,	,	PUNCT
ejpam-4396	32	38	there	there	PRON
ejpam-4396	32	39	exists	exist	VERB
ejpam-4396	32	40	a	a	DET
ejpam-4396	32	41	unique	unique	ADJ
ejpam-4396	32	42	element	element	NOUN
ejpam-4396	32	43	e(g	e(g	NOUN
ejpam-4396	32	44	)	)	PUNCT
ejpam-4396	32	45	such	such	ADJ
ejpam-4396	32	46	that	that	SCONJ
ejpam-4396	32	47	g	g	PROPN
ejpam-4396	32	48	∗	∗	PROPN
ejpam-4396	32	49	e(g	e(g	PROPN
ejpam-4396	32	50	)	)	PUNCT
ejpam-4396	33	1	=	=	SYM
ejpam-4396	33	2	g	g	PROPN
ejpam-4396	33	3	=	=	SYM
ejpam-4396	33	4	e(g	e(g	PROPN
ejpam-4396	33	5	)	)	PUNCT
ejpam-4396	33	6	∗	∗	VERB
ejpam-4396	33	7	g	g	NOUN
ejpam-4396	33	8	;	;	PUNCT
ejpam-4396	33	9	and	and	CCONJ
ejpam-4396	33	10	,	,	PUNCT
ejpam-4396	33	11	(	(	PUNCT
ejpam-4396	33	12	m3	m3	PROPN
ejpam-4396	33	13	)	)	PUNCT
ejpam-4396	33	14	for	for	ADP
ejpam-4396	33	15	each	each	DET
ejpam-4396	33	16	g	g	PROPN
ejpam-4396	33	17	∈	∈	PROPN
ejpam-4396	33	18	g	g	NOUN
ejpam-4396	33	19	,	,	PUNCT
ejpam-4396	33	20	there	there	PRON
ejpam-4396	33	21	exists	exist	VERB
ejpam-4396	33	22	an	an	DET
ejpam-4396	33	23	element	element	NOUN
ejpam-4396	33	24	h	h	NOUN
ejpam-4396	33	25	∈	∈	PROPN
ejpam-4396	33	26	g	g	PROPN
ejpam-4396	33	27	such	such	ADJ
ejpam-4396	33	28	that	that	SCONJ
ejpam-4396	33	29	g	g	PROPN
ejpam-4396	33	30	∗	∗	NOUN
ejpam-4396	33	31	h	h	NOUN
ejpam-4396	34	1	=	=	NOUN
ejpam-4396	34	2	h	h	PROPN
ejpam-4396	34	3	∗	∗	NOUN
ejpam-4396	34	4	g	g	PROPN
ejpam-4396	34	5	=	=	PROPN
ejpam-4396	34	6	e(g	e(g	PROPN
ejpam-4396	34	7	)	)	PUNCT
ejpam-4396	34	8	.	.	PUNCT
ejpam-4396	35	1	hereafter	hereafter	ADV
ejpam-4396	35	2	,	,	PUNCT
ejpam-4396	35	3	please	please	INTJ
ejpam-4396	35	4	refer	refer	VERB
ejpam-4396	35	5	to	to	ADP
ejpam-4396	35	6	[	[	X
ejpam-4396	35	7	3	3	X
ejpam-4396	35	8	]	]	PUNCT
ejpam-4396	35	9	for	for	ADP
ejpam-4396	35	10	the	the	DET
ejpam-4396	35	11	other	other	ADJ
ejpam-4396	35	12	concepts	concept	NOUN
ejpam-4396	35	13	.	.	PUNCT
ejpam-4396	36	1	this	this	DET
ejpam-4396	36	2	paper	paper	NOUN
ejpam-4396	36	3	is	be	AUX
ejpam-4396	36	4	a	a	DET
ejpam-4396	36	5	sequel	sequel	NOUN
ejpam-4396	36	6	of	of	ADP
ejpam-4396	36	7	a	a	DET
ejpam-4396	36	8	previously	previously	ADV
ejpam-4396	36	9	published	publish	VERB
ejpam-4396	36	10	study	study	NOUN
ejpam-4396	36	11	[	[	X
ejpam-4396	36	12	1	1	X
ejpam-4396	36	13	]	]	PUNCT
ejpam-4396	36	14	where	where	SCONJ
ejpam-4396	36	15	a	a	DET
ejpam-4396	36	16	new	new	ADJ
ejpam-4396	36	17	algebraic	algebraic	ADJ
ejpam-4396	36	18	structure	structure	NOUN
ejpam-4396	36	19	called	call	VERB
ejpam-4396	36	20	g	g	NOUN
ejpam-4396	36	21	-	-	PUNCT
ejpam-4396	36	22	group	group	NOUN
ejpam-4396	36	23	was	be	AUX
ejpam-4396	36	24	introduced	introduce	VERB
ejpam-4396	36	25	.	.	PUNCT
ejpam-4396	37	1	additional	additional	ADJ
ejpam-4396	37	2	properties	property	NOUN
ejpam-4396	37	3	of	of	ADP
ejpam-4396	37	4	such	such	ADJ
ejpam-4396	37	5	structure	structure	NOUN
ejpam-4396	37	6	is	be	AUX
ejpam-4396	37	7	presented	present	VERB
ejpam-4396	37	8	and	and	CCONJ
ejpam-4396	37	9	shown	show	VERB
ejpam-4396	37	10	in	in	ADP
ejpam-4396	37	11	this	this	DET
ejpam-4396	37	12	paper	paper	NOUN
ejpam-4396	37	13	.	.	PUNCT
ejpam-4396	38	1	in	in	ADP
ejpam-4396	38	2	the	the	DET
ejpam-4396	38	3	early	early	ADJ
ejpam-4396	38	4	twentieth	twentieth	ADJ
ejpam-4396	38	5	century	century	NOUN
ejpam-4396	38	6	,	,	PUNCT
ejpam-4396	38	7	algebra	algebra	NOUN
ejpam-4396	38	8	had	have	AUX
ejpam-4396	38	9	evolved	evolve	VERB
ejpam-4396	38	10	into	into	ADP
ejpam-4396	38	11	a	a	DET
ejpam-4396	38	12	study	study	NOUN
ejpam-4396	38	13	of	of	ADP
ejpam-4396	38	14	axiomatic	axiomatic	ADJ
ejpam-4396	38	15	systems	system	NOUN
ejpam-4396	38	16	.	.	PUNCT
ejpam-4396	39	1	it	it	PRON
ejpam-4396	39	2	was	be	AUX
ejpam-4396	39	3	then	then	ADV
ejpam-4396	39	4	referred	refer	VERB
ejpam-4396	39	5	to	to	ADP
ejpam-4396	39	6	as	as	ADP
ejpam-4396	39	7	abstract	abstract	ADJ
ejpam-4396	39	8	algebra	algebra	NOUN
ejpam-4396	39	9	[	[	X
ejpam-4396	39	10	5	5	NUM
ejpam-4396	39	11	]	]	PUNCT
ejpam-4396	39	12	.	.	PUNCT
ejpam-4396	40	1	since	since	SCONJ
ejpam-4396	40	2	then	then	ADV
ejpam-4396	40	3	,	,	PUNCT
ejpam-4396	40	4	mathematicians	mathematician	NOUN
ejpam-4396	40	5	have	have	AUX
ejpam-4396	40	6	introduced	introduce	VERB
ejpam-4396	40	7	and	and	CCONJ
ejpam-4396	40	8	explored	explore	VERB
ejpam-4396	40	9	various	various	ADJ
ejpam-4396	40	10	algebraic	algebraic	ADJ
ejpam-4396	40	11	structures	structure	NOUN
ejpam-4396	40	12	.	.	PUNCT
ejpam-4396	41	1	some	some	PRON
ejpam-4396	41	2	were	be	AUX
ejpam-4396	41	3	found	find	VERB
ejpam-4396	41	4	related	related	ADJ
ejpam-4396	41	5	to	to	ADP
ejpam-4396	41	6	another	another	PRON
ejpam-4396	41	7	and	and	CCONJ
ejpam-4396	41	8	others	other	NOUN
ejpam-4396	41	9	were	be	AUX
ejpam-4396	41	10	found	find	VERB
ejpam-4396	41	11	to	to	PART
ejpam-4396	41	12	be	be	AUX
ejpam-4396	41	13	entirely	entirely	ADV
ejpam-4396	41	14	different	different	ADJ
ejpam-4396	41	15	.	.	PUNCT
ejpam-4396	42	1	one	one	NUM
ejpam-4396	42	2	particular	particular	ADJ
ejpam-4396	42	3	concept	concept	NOUN
ejpam-4396	42	4	that	that	PRON
ejpam-4396	42	5	captured	capture	VERB
ejpam-4396	42	6	the	the	DET
ejpam-4396	42	7	attention	attention	NOUN
ejpam-4396	42	8	of	of	ADP
ejpam-4396	42	9	many	many	ADJ
ejpam-4396	42	10	researchers	researcher	NOUN
ejpam-4396	42	11	is	be	AUX
ejpam-4396	42	12	that	that	PRON
ejpam-4396	42	13	of	of	ADP
ejpam-4396	42	14	groups	group	NOUN
ejpam-4396	42	15	.	.	PUNCT
ejpam-4396	43	1	several	several	ADJ
ejpam-4396	43	2	group	group	NOUN
ejpam-4396	43	3	-	-	PUNCT
ejpam-4396	43	4	related	relate	VERB
ejpam-4396	43	5	structures	structure	NOUN
ejpam-4396	43	6	such	such	ADJ
ejpam-4396	43	7	as	as	ADP
ejpam-4396	43	8	quasigroups	quasigroup	NOUN
ejpam-4396	43	9	,	,	PUNCT
ejpam-4396	43	10	generalized	generalized	ADJ
ejpam-4396	43	11	groups	group	NOUN
ejpam-4396	43	12	and	and	CCONJ
ejpam-4396	43	13	similar	similar	ADJ
ejpam-4396	43	14	structures	structure	NOUN
ejpam-4396	43	15	became	become	VERB
ejpam-4396	43	16	the	the	DET
ejpam-4396	43	17	favorite	favorite	ADJ
ejpam-4396	43	18	topic	topic	NOUN
ejpam-4396	43	19	of	of	ADP
ejpam-4396	43	20	algebra	algebra	NOUN
ejpam-4396	43	21	enthusiasts	enthusiast	NOUN
ejpam-4396	43	22	[	[	X
ejpam-4396	43	23	6	6	NUM
ejpam-4396	43	24	]	]	PUNCT
ejpam-4396	43	25	,	,	PUNCT
ejpam-4396	43	26	[	[	X
ejpam-4396	43	27	2	2	NUM
ejpam-4396	43	28	]	]	PUNCT
ejpam-4396	43	29	,	,	PUNCT
ejpam-4396	43	30	[	[	X
ejpam-4396	43	31	7	7	NUM
ejpam-4396	43	32	]	]	PUNCT
ejpam-4396	43	33	.	.	PUNCT
ejpam-4396	44	1	findings	finding	NOUN
ejpam-4396	44	2	from	from	ADP
ejpam-4396	44	3	these	these	DET
ejpam-4396	44	4	studies	study	NOUN
ejpam-4396	44	5	were	be	AUX
ejpam-4396	44	6	found	find	VERB
ejpam-4396	44	7	to	to	PART
ejpam-4396	44	8	be	be	AUX
ejpam-4396	44	9	applicable	applicable	ADJ
ejpam-4396	44	10	in	in	ADP
ejpam-4396	44	11	other	other	ADJ
ejpam-4396	44	12	branches	branch	NOUN
ejpam-4396	44	13	of	of	ADP
ejpam-4396	44	14	mathematics	mathematic	NOUN
ejpam-4396	44	15	such	such	ADJ
ejpam-4396	44	16	as	as	ADP
ejpam-4396	44	17	number	number	NOUN
ejpam-4396	44	18	theory	theory	NOUN
ejpam-4396	44	19	,	,	PUNCT
ejpam-4396	44	20	geometry	geometry	NOUN
ejpam-4396	44	21	,	,	PUNCT
ejpam-4396	44	22	analysis	analysis	NOUN
ejpam-4396	44	23	[	[	X
ejpam-4396	44	24	4	4	NUM
ejpam-4396	44	25	]	]	PUNCT
ejpam-4396	44	26	,	,	PUNCT
ejpam-4396	44	27	computer	computer	NOUN
ejpam-4396	44	28	science	science	NOUN
ejpam-4396	45	1	[	[	X
ejpam-4396	45	2	1	1	NUM
ejpam-4396	45	3	]	]	PUNCT
ejpam-4396	45	4	,	,	PUNCT
ejpam-4396	45	5	etc	etc	X
ejpam-4396	45	6	.	.	X
ejpam-4396	45	7	unlike	unlike	ADP
ejpam-4396	45	8	in	in	ADP
ejpam-4396	45	9	groups	group	NOUN
ejpam-4396	45	10	,	,	PUNCT
ejpam-4396	45	11	distinct	distinct	ADJ
ejpam-4396	45	12	elements	element	NOUN
ejpam-4396	45	13	of	of	ADP
ejpam-4396	45	14	a	a	DET
ejpam-4396	45	15	g	g	NOUN
ejpam-4396	45	16	-	-	PUNCT
ejpam-4396	45	17	group	group	NOUN
ejpam-4396	45	18	may	may	AUX
ejpam-4396	45	19	have	have	VERB
ejpam-4396	45	20	different	different	ADJ
ejpam-4396	45	21	identity	identity	NOUN
ejpam-4396	45	22	elements	element	NOUN
ejpam-4396	45	23	.	.	PUNCT
ejpam-4396	46	1	also	also	ADV
ejpam-4396	46	2	,	,	PUNCT
ejpam-4396	46	3	the	the	DET
ejpam-4396	46	4	identity	identity	NOUN
ejpam-4396	46	5	element	element	NOUN
ejpam-4396	46	6	as	as	ADV
ejpam-4396	46	7	well	well	ADV
ejpam-4396	46	8	as	as	ADP
ejpam-4396	46	9	the	the	DET
ejpam-4396	46	10	inverse	inverse	NOUN
ejpam-4396	46	11	may	may	AUX
ejpam-4396	46	12	not	not	PART
ejpam-4396	46	13	be	be	AUX
ejpam-4396	46	14	unique	unique	ADJ
ejpam-4396	46	15	.	.	PUNCT
ejpam-4396	47	1	a	a	DET
ejpam-4396	47	2	g	g	NOUN
ejpam-4396	47	3	-	-	PUNCT
ejpam-4396	47	4	group	group	NOUN
ejpam-4396	47	5	is	be	AUX
ejpam-4396	47	6	generally	generally	ADV
ejpam-4396	47	7	not	not	PART
ejpam-4396	47	8	a	a	DET
ejpam-4396	47	9	group	group	NOUN
ejpam-4396	47	10	,	,	PUNCT
ejpam-4396	47	11	but	but	CCONJ
ejpam-4396	47	12	groups	group	NOUN
ejpam-4396	47	13	are	be	AUX
ejpam-4396	47	14	necessarily	necessarily	ADV
ejpam-4396	47	15	g	g	NOUN
ejpam-4396	47	16	-	-	PUNCT
ejpam-4396	47	17	groups	group	NOUN
ejpam-4396	47	18	.	.	PUNCT
ejpam-4396	48	1	distinctions	distinction	NOUN
ejpam-4396	48	2	of	of	ADP
ejpam-4396	48	3	g	g	NOUN
ejpam-4396	48	4	-	-	PUNCT
ejpam-4396	48	5	groups	group	NOUN
ejpam-4396	48	6	from	from	ADP
ejpam-4396	48	7	other	other	ADJ
ejpam-4396	48	8	group	group	NOUN
ejpam-4396	48	9	-	-	PUNCT
ejpam-4396	48	10	like	like	ADJ
ejpam-4396	48	11	structures	structure	NOUN
ejpam-4396	48	12	like	like	ADP
ejpam-4396	48	13	generalized	generalized	ADJ
ejpam-4396	48	14	group	group	NOUN
ejpam-4396	48	15	and	and	CCONJ
ejpam-4396	48	16	e	e	NOUN
ejpam-4396	48	17	-	-	NOUN
ejpam-4396	48	18	group	group	NOUN
ejpam-4396	48	19	are	be	AUX
ejpam-4396	48	20	established	establish	VERB
ejpam-4396	48	21	in	in	ADP
ejpam-4396	48	22	[	[	X
ejpam-4396	48	23	1	1	NUM
ejpam-4396	48	24	]	]	PUNCT
ejpam-4396	48	25	.	.	PUNCT
ejpam-4396	49	1	ubat	ubat	PROPN
ejpam-4396	49	2	et	et	PROPN
ejpam-4396	49	3	al	al	PROPN
ejpam-4396	49	4	.	.	PUNCT
ejpam-4396	50	1	[	[	X
ejpam-4396	50	2	1	1	X
ejpam-4396	50	3	]	]	PUNCT
ejpam-4396	50	4	presented	present	VERB
ejpam-4396	50	5	figure	figure	NOUN
ejpam-4396	50	6	1	1	NUM
ejpam-4396	50	7	,	,	PUNCT
ejpam-4396	50	8	which	which	PRON
ejpam-4396	50	9	briefly	briefly	NOUN
ejpam-4396	50	10	summarizes	summarize	VERB
ejpam-4396	50	11	the	the	DET
ejpam-4396	50	12	relationship	relationship	NOUN
ejpam-4396	50	13	of	of	ADP
ejpam-4396	50	14	the	the	DET
ejpam-4396	50	15	j.	j.	PROPN
ejpam-4396	50	16	caraquil	caraquil	PROPN
ejpam-4396	50	17	,	,	PUNCT
ejpam-4396	50	18	m.	m.	PROPN
ejpam-4396	50	19	baldado	baldado	PROPN
ejpam-4396	50	20	jr	jr	PROPN
ejpam-4396	50	21	.	.	PROPN
ejpam-4396	50	22	/	/	SYM
ejpam-4396	50	23	eur	eur	PROPN
ejpam-4396	50	24	.	.	PUNCT
ejpam-4396	51	1	j.	j.	PROPN
ejpam-4396	51	2	pure	pure	PROPN
ejpam-4396	51	3	appl	appl	PROPN
ejpam-4396	51	4	.	.	PROPN
ejpam-4396	51	5	math	math	PROPN
ejpam-4396	51	6	,	,	PUNCT
ejpam-4396	51	7	15	15	NUM
ejpam-4396	51	8	(	(	PUNCT
ejpam-4396	51	9	3	3	NUM
ejpam-4396	51	10	)	)	PUNCT
ejpam-4396	51	11	(	(	PUNCT
ejpam-4396	51	12	2022	2022	NUM
ejpam-4396	51	13	)	)	PUNCT
ejpam-4396	51	14	,	,	PUNCT
ejpam-4396	51	15	887	887	NUM
ejpam-4396	51	16	-	-	SYM
ejpam-4396	51	17	896	896	NUM
ejpam-4396	51	18	889	889	NUM
ejpam-4396	51	19	×	×	NOUN
ejpam-4396	51	20	0	0	NUM
ejpam-4396	51	21	1	1	NUM
ejpam-4396	51	22	0	0	NUM
ejpam-4396	51	23	0	0	NUM
ejpam-4396	51	24	0	0	NUM
ejpam-4396	51	25	1	1	NUM
ejpam-4396	51	26	0	0	NUM
ejpam-4396	51	27	1	1	NUM
ejpam-4396	51	28	table	table	NOUN
ejpam-4396	51	29	3	3	NUM
ejpam-4396	51	30	:	:	PUNCT
ejpam-4396	51	31	the	the	DET
ejpam-4396	51	32	g	g	PROPN
ejpam-4396	51	33	-	-	PUNCT
ejpam-4396	51	34	group	group	NOUN
ejpam-4396	51	35	{	{	PUNCT
ejpam-4396	51	36	0	0	NUM
ejpam-4396	51	37	,	,	PUNCT
ejpam-4396	51	38	1	1	NUM
ejpam-4396	51	39	}	}	PUNCT
ejpam-4396	51	40	different	different	ADJ
ejpam-4396	51	41	algebraic	algebraic	ADJ
ejpam-4396	51	42	structure	structure	NOUN
ejpam-4396	51	43	.	.	PUNCT
ejpam-4396	52	1	solid	solid	ADJ
ejpam-4396	52	2	arcs	arc	NOUN
ejpam-4396	52	3	represent	represent	VERB
ejpam-4396	52	4	the	the	DET
ejpam-4396	52	5	fact	fact	NOUN
ejpam-4396	52	6	that	that	SCONJ
ejpam-4396	52	7	the	the	DET
ejpam-4396	52	8	family	family	NOUN
ejpam-4396	52	9	in	in	ADP
ejpam-4396	52	10	the	the	DET
ejpam-4396	52	11	tail	tail	NOUN
ejpam-4396	52	12	is	be	AUX
ejpam-4396	52	13	a	a	DET
ejpam-4396	52	14	subset	subset	NOUN
ejpam-4396	52	15	of	of	ADP
ejpam-4396	52	16	the	the	DET
ejpam-4396	52	17	one	one	NOUN
ejpam-4396	52	18	in	in	ADP
ejpam-4396	52	19	the	the	DET
ejpam-4396	52	20	head	head	NOUN
ejpam-4396	52	21	.	.	PUNCT
ejpam-4396	53	1	while	while	SCONJ
ejpam-4396	53	2	dashed	dash	VERB
ejpam-4396	53	3	arcs	arc	NOUN
ejpam-4396	53	4	represent	represent	VERB
ejpam-4396	53	5	the	the	DET
ejpam-4396	53	6	idea	idea	NOUN
ejpam-4396	53	7	’	'	PUNCT
ejpam-4396	53	8	can	can	AUX
ejpam-4396	53	9	be	be	AUX
ejpam-4396	53	10	made	make	VERB
ejpam-4396	53	11	’	'	PUNCT
ejpam-4396	53	12	.	.	PUNCT
ejpam-4396	54	1	for	for	ADP
ejpam-4396	54	2	example	example	NOUN
ejpam-4396	54	3	,	,	PUNCT
ejpam-4396	54	4	a	a	DET
ejpam-4396	54	5	dashed	dash	VERB
ejpam-4396	54	6	line	line	NOUN
ejpam-4396	54	7	is	be	AUX
ejpam-4396	54	8	drawn	draw	VERB
ejpam-4396	54	9	from	from	ADP
ejpam-4396	54	10	the	the	DET
ejpam-4396	54	11	family	family	NOUN
ejpam-4396	54	12	of	of	ADP
ejpam-4396	54	13	ubat	ubat	NOUN
ejpam-4396	54	14	-	-	PUNCT
ejpam-4396	54	15	spaces	space	NOUN
ejpam-4396	54	16	to	to	ADP
ejpam-4396	54	17	the	the	DET
ejpam-4396	54	18	family	family	NOUN
ejpam-4396	54	19	of	of	ADP
ejpam-4396	54	20	e	e	NOUN
ejpam-4396	54	21	-	-	NOUN
ejpam-4396	54	22	groups	group	NOUN
ejpam-4396	54	23	since	since	SCONJ
ejpam-4396	54	24	,	,	PUNCT
ejpam-4396	54	25	although	although	SCONJ
ejpam-4396	54	26	ubat	ubat	ADJ
ejpam-4396	54	27	-	-	PUNCT
ejpam-4396	54	28	spaces	space	NOUN
ejpam-4396	54	29	(	(	PUNCT
ejpam-4396	54	30	g	g	NOUN
ejpam-4396	54	31	,	,	PUNCT
ejpam-4396	54	32	∗	∗	NOUN
ejpam-4396	54	33	)	)	PUNCT
ejpam-4396	54	34	and	and	CCONJ
ejpam-4396	54	35	e	e	NOUN
ejpam-4396	54	36	-	-	NOUN
ejpam-4396	54	37	groups	group	NOUN
ejpam-4396	54	38	are	be	AUX
ejpam-4396	54	39	non	non	X
ejpam-4396	54	40	comparable	comparable	ADJ
ejpam-4396	54	41	structures	structure	NOUN
ejpam-4396	54	42	,	,	PUNCT
ejpam-4396	54	43	but	but	CCONJ
ejpam-4396	54	44	a	a	DET
ejpam-4396	54	45	subset	subset	NOUN
ejpam-4396	54	46	a	a	PRON
ejpam-4396	54	47	from	from	ADP
ejpam-4396	54	48	g	g	NOUN
ejpam-4396	54	49	can	can	AUX
ejpam-4396	54	50	be	be	AUX
ejpam-4396	54	51	chosen	choose	VERB
ejpam-4396	54	52	,	,	PUNCT
ejpam-4396	54	53	so	so	SCONJ
ejpam-4396	54	54	that	that	SCONJ
ejpam-4396	54	55	(	(	PUNCT
ejpam-4396	54	56	g	g	NOUN
ejpam-4396	54	57	;	;	PUNCT
ejpam-4396	54	58	∗;a	∗;a	NOUN
ejpam-4396	54	59	)	)	PUNCT
ejpam-4396	54	60	is	be	AUX
ejpam-4396	54	61	an	an	DET
ejpam-4396	54	62	e	e	NOUN
ejpam-4396	54	63	-	-	NOUN
ejpam-4396	54	64	group	group	NOUN
ejpam-4396	54	65	.	.	PUNCT
ejpam-4396	55	1	3	3	NUM
ejpam-4396	55	2	=	=	SYM
ejpam-4396	55	3	2.2pt	2.2pt	NUM
ejpam-4396	55	4	=	=	SYM
ejpam-4396	55	5	2.1pt(38	2.1pt(38	NOUN
ejpam-4396	55	6	,	,	PUNCT
ejpam-4396	55	7	0)(68	0)(68	NUM
ejpam-4396	55	8	,	,	PUNCT
ejpam-4396	55	9	0)(86	0)(86	PROPN
ejpam-4396	55	10	,	,	PUNCT
ejpam-4396	55	11	4)(62	4)(62	NUM
ejpam-4396	55	12	,	,	PUNCT
ejpam-4396	55	13	26)(−30	26)(−30	NUM
ejpam-4396	55	14	,	,	PUNCT
ejpam-4396	55	15	4)(0	4)(0	NUM
ejpam-4396	55	16	,	,	PUNCT
ejpam-4396	55	17	26)(27	26)(27	NUM
ejpam-4396	55	18	,	,	PUNCT
ejpam-4396	55	19	4)(3	4)(3	NUM
ejpam-4396	55	20	,	,	PUNCT
ejpam-4396	55	21	26)(49	26)(49	NUM
ejpam-4396	55	22	,	,	PUNCT
ejpam-4396	55	23	30)(11	30)(11	NUM
ejpam-4396	55	24	,	,	PUNCT
ejpam-4396	55	25	30)(83	30)(83	NUM
ejpam-4396	55	26	,	,	PUNCT
ejpam-4396	55	27	4)(5	4)(5	NUM
ejpam-4396	55	28	,	,	PUNCT
ejpam-4396	55	29	26	26	NUM
ejpam-4396	55	30	)	)	PUNCT
ejpam-4396	55	31	e−groupat030ubat−	e−groupat030ubat−	NOUN
ejpam-4396	55	32	spaceat−	spaceat−	NOUN
ejpam-4396	55	33	300groupat300g−groupat6030generalizedgroupat900	300groupat300g−groupat6030generalizedgroupat900	ADJ
ejpam-4396	55	34	figure	figure	NOUN
ejpam-4396	55	35	1	1	NUM
ejpam-4396	55	36	:	:	PUNCT
ejpam-4396	55	37	relationship	relationship	NOUN
ejpam-4396	55	38	in	in	ADP
ejpam-4396	55	39	terms	term	NOUN
ejpam-4396	55	40	of	of	ADP
ejpam-4396	55	41	set	set	VERB
ejpam-4396	55	42	theoretic	theoretic	ADJ
ejpam-4396	55	43	inclusion	inclusion	NOUN
ejpam-4396	55	44	of	of	ADP
ejpam-4396	55	45	the	the	DET
ejpam-4396	55	46	classes	class	NOUN
ejpam-4396	55	47	of	of	ADP
ejpam-4396	55	48	groups	group	NOUN
ejpam-4396	55	49	,	,	PUNCT
ejpam-4396	55	50	g	g	NOUN
ejpam-4396	55	51	-	-	PUNCT
ejpam-4396	55	52	groups	group	NOUN
ejpam-4396	55	53	,	,	PUNCT
ejpam-4396	55	54	e	e	NOUN
ejpam-4396	55	55	-	-	NOUN
ejpam-4396	55	56	groups	group	NOUN
ejpam-4396	55	57	,	,	PUNCT
ejpam-4396	55	58	generalized	generalized	ADJ
ejpam-4396	55	59	groups	group	NOUN
ejpam-4396	55	60	and	and	CCONJ
ejpam-4396	55	61	ubat	ubat	ADJ
ejpam-4396	55	62	-	-	PUNCT
ejpam-4396	55	63	spaces	space	VERB
ejpam-4396	55	64	the	the	DET
ejpam-4396	55	65	structure	structure	NOUN
ejpam-4396	55	66	g	g	PROPN
ejpam-4396	55	67	-	-	PUNCT
ejpam-4396	55	68	group	group	NOUN
ejpam-4396	55	69	may	may	AUX
ejpam-4396	55	70	have	have	VERB
ejpam-4396	55	71	important	important	ADJ
ejpam-4396	55	72	applications	application	NOUN
ejpam-4396	55	73	in	in	ADP
ejpam-4396	55	74	microprocessor	microprocessor	NOUN
ejpam-4396	55	75	design	design	NOUN
ejpam-4396	55	76	.	.	PUNCT
ejpam-4396	56	1	specifically	specifically	ADV
ejpam-4396	56	2	,	,	PUNCT
ejpam-4396	56	3	it	it	PRON
ejpam-4396	56	4	can	can	AUX
ejpam-4396	56	5	be	be	AUX
ejpam-4396	56	6	used	use	VERB
ejpam-4396	56	7	to	to	PART
ejpam-4396	56	8	minimize	minimize	VERB
ejpam-4396	56	9	digital	digital	ADJ
ejpam-4396	56	10	circuits	circuit	NOUN
ejpam-4396	56	11	which	which	PRON
ejpam-4396	56	12	uses	use	VERB
ejpam-4396	56	13	and	and	CCONJ
ejpam-4396	56	14	gates	gate	NOUN
ejpam-4396	56	15	only	only	ADV
ejpam-4396	56	16	.	.	PUNCT
ejpam-4396	57	1	for	for	ADP
ejpam-4396	57	2	example	example	NOUN
ejpam-4396	57	3	,	,	PUNCT
ejpam-4396	57	4	consider	consider	VERB
ejpam-4396	57	5	the	the	DET
ejpam-4396	57	6	digital	digital	ADJ
ejpam-4396	57	7	circuit	circuit	NOUN
ejpam-4396	57	8	with	with	ADP
ejpam-4396	57	9	three	three	NUM
ejpam-4396	57	10	inputs	input	NOUN
ejpam-4396	57	11	,	,	PUNCT
ejpam-4396	57	12	a	a	DET
ejpam-4396	57	13	,	,	PUNCT
ejpam-4396	57	14	b	b	NOUN
ejpam-4396	57	15	,	,	PUNCT
ejpam-4396	57	16	and	and	CCONJ
ejpam-4396	57	17	c	c	X
ejpam-4396	57	18	,	,	PUNCT
ejpam-4396	57	19	given	give	VERB
ejpam-4396	57	20	by	by	ADP
ejpam-4396	57	21	(	(	PUNCT
ejpam-4396	57	22	a∨b)∨(a∨c	a∨b)∨(a∨c	NOUN
ejpam-4396	57	23	)	)	PUNCT
ejpam-4396	57	24	.	.	PUNCT
ejpam-4396	58	1	by	by	ADP
ejpam-4396	58	2	inspection	inspection	NOUN
ejpam-4396	58	3	,	,	PUNCT
ejpam-4396	58	4	the	the	DET
ejpam-4396	58	5	expression	expression	NOUN
ejpam-4396	58	6	(	(	PUNCT
ejpam-4396	58	7	a	a	DET
ejpam-4396	58	8	∨	∨	NUM
ejpam-4396	58	9	b	b	NOUN
ejpam-4396	58	10	)	)	PUNCT
ejpam-4396	58	11	∨	∨	NOUN
ejpam-4396	58	12	(	(	PUNCT
ejpam-4396	58	13	a	a	DET
ejpam-4396	58	14	∨	∨	NUM
ejpam-4396	58	15	c	c	NOUN
ejpam-4396	58	16	)	)	PUNCT
ejpam-4396	58	17	suggest	suggest	VERB
ejpam-4396	58	18	that	that	SCONJ
ejpam-4396	58	19	a	a	DET
ejpam-4396	58	20	digital	digital	ADJ
ejpam-4396	58	21	circuit	circuit	NOUN
ejpam-4396	58	22	needs	need	VERB
ejpam-4396	58	23	three	three	NUM
ejpam-4396	58	24	and	and	CCONJ
ejpam-4396	58	25	gates	gate	NOUN
ejpam-4396	58	26	to	to	PART
ejpam-4396	58	27	give	give	VERB
ejpam-4396	58	28	the	the	DET
ejpam-4396	58	29	desired	desire	VERB
ejpam-4396	58	30	output	output	NOUN
ejpam-4396	58	31	.	.	PUNCT
ejpam-4396	59	1	however	however	ADV
ejpam-4396	59	2	,	,	PUNCT
ejpam-4396	59	3	using	use	VERB
ejpam-4396	59	4	some	some	DET
ejpam-4396	59	5	properties	property	NOUN
ejpam-4396	59	6	of	of	ADP
ejpam-4396	59	7	the	the	DET
ejpam-4396	59	8	g	g	NOUN
ejpam-4396	59	9	-	-	PUNCT
ejpam-4396	59	10	group	group	NOUN
ejpam-4396	59	11	(	(	PUNCT
ejpam-4396	59	12	z2	z2	PROPN
ejpam-4396	59	13	,	,	PUNCT
ejpam-4396	59	14	·	·	PUNCT
ejpam-4396	59	15	)	)	PUNCT
ejpam-4396	59	16	,	,	PUNCT
ejpam-4396	59	17	the	the	DET
ejpam-4396	59	18	circuit	circuit	NOUN
ejpam-4396	59	19	can	can	AUX
ejpam-4396	59	20	be	be	AUX
ejpam-4396	59	21	minimized	minimize	VERB
ejpam-4396	59	22	as	as	SCONJ
ejpam-4396	59	23	follows	follow	VERB
ejpam-4396	59	24	.	.	PUNCT
ejpam-4396	60	1	identifying	identify	VERB
ejpam-4396	60	2	·	·	PUNCT
ejpam-4396	60	3	with	with	ADP
ejpam-4396	60	4	∨	∨	NUM
ejpam-4396	60	5	,	,	PUNCT
ejpam-4396	60	6	we	we	PRON
ejpam-4396	60	7	have	have	AUX
ejpam-4396	60	8	(	(	PUNCT
ejpam-4396	60	9	a∨b)∨(a∨c	a∨b)∨(a∨c	NOUN
ejpam-4396	60	10	)	)	PUNCT
ejpam-4396	60	11	=	=	PUNCT
ejpam-4396	60	12	(	(	PUNCT
ejpam-4396	60	13	a	a	DET
ejpam-4396	60	14	·	·	SYM
ejpam-4396	60	15	b	b	NOUN
ejpam-4396	60	16	)	)	PUNCT
ejpam-4396	60	17	·	·	PUNCT
ejpam-4396	60	18	(	(	PUNCT
ejpam-4396	60	19	a	a	DET
ejpam-4396	60	20	·	·	SYM
ejpam-4396	60	21	c	c	NOUN
ejpam-4396	60	22	)	)	PUNCT
ejpam-4396	60	23	=	=	SYM
ejpam-4396	61	1	[	[	X
ejpam-4396	61	2	(	(	PUNCT
ejpam-4396	61	3	a	a	DET
ejpam-4396	61	4	·	·	SYM
ejpam-4396	61	5	b	b	NOUN
ejpam-4396	61	6	)	)	PUNCT
ejpam-4396	61	7	·	·	PUNCT
ejpam-4396	61	8	a	a	X
ejpam-4396	61	9	]	]	X
ejpam-4396	61	10	·	·	PUNCT
ejpam-4396	61	11	c	c	X
ejpam-4396	61	12	=	=	PUNCT
ejpam-4396	62	1	[	[	X
ejpam-4396	62	2	a	a	X
ejpam-4396	62	3	·	·	PUNCT
ejpam-4396	62	4	(	(	PUNCT
ejpam-4396	62	5	b	b	X
ejpam-4396	62	6	·	·	SYM
ejpam-4396	62	7	a	a	NOUN
ejpam-4396	62	8	)	)	PUNCT
ejpam-4396	62	9	]	]	PUNCT
ejpam-4396	62	10	·	·	PUNCT
ejpam-4396	62	11	c	c	X
ejpam-4396	62	12	=	=	PUNCT
ejpam-4396	63	1	[	[	X
ejpam-4396	63	2	a	a	X
ejpam-4396	63	3	·	·	PUNCT
ejpam-4396	63	4	(	(	PUNCT
ejpam-4396	63	5	a	a	DET
ejpam-4396	63	6	·	·	SYM
ejpam-4396	63	7	b	b	NOUN
ejpam-4396	63	8	)	)	PUNCT
ejpam-4396	63	9	]	]	PUNCT
ejpam-4396	63	10	·	·	PUNCT
ejpam-4396	63	11	c	c	X
ejpam-4396	63	12	=	=	PUNCT
ejpam-4396	64	1	[	[	X
ejpam-4396	64	2	(	(	PUNCT
ejpam-4396	64	3	a	a	PRON
ejpam-4396	64	4	·	·	SYM
ejpam-4396	64	5	a	a	NOUN
ejpam-4396	64	6	)	)	PUNCT
ejpam-4396	64	7	·	·	PUNCT
ejpam-4396	64	8	b	b	NOUN
ejpam-4396	64	9	]	]	PUNCT
ejpam-4396	64	10	·	·	PUNCT
ejpam-4396	64	11	c	c	X
ejpam-4396	64	12	=	=	SYM
ejpam-4396	64	13	(	(	PUNCT
ejpam-4396	64	14	a	a	DET
ejpam-4396	64	15	·	·	SYM
ejpam-4396	64	16	b	b	X
ejpam-4396	64	17	)	)	PUNCT
ejpam-4396	64	18	·	·	PUNCT
ejpam-4396	65	1	c	c	X
ejpam-4396	65	2	=	=	SYM
ejpam-4396	65	3	(	(	PUNCT
ejpam-4396	65	4	a	a	DET
ejpam-4396	65	5	∨	∨	NUM
ejpam-4396	65	6	b	b	NOUN
ejpam-4396	65	7	)	)	PUNCT
ejpam-4396	65	8	∨	∨	PROPN
ejpam-4396	65	9	c.	c.	NOUN
ejpam-4396	65	10	note	note	VERB
ejpam-4396	65	11	that	that	SCONJ
ejpam-4396	65	12	the	the	DET
ejpam-4396	65	13	expression	expression	NOUN
ejpam-4396	65	14	(	(	PUNCT
ejpam-4396	65	15	a	a	DET
ejpam-4396	65	16	∨	∨	NUM
ejpam-4396	65	17	b	b	NOUN
ejpam-4396	65	18	)	)	PUNCT
ejpam-4396	65	19	∨	∨	NUM
ejpam-4396	65	20	c	c	PROPN
ejpam-4396	65	21	uses	use	VERB
ejpam-4396	65	22	two	two	NUM
ejpam-4396	65	23	and	and	CCONJ
ejpam-4396	65	24	gates	gate	NOUN
ejpam-4396	65	25	only	only	ADV
ejpam-4396	65	26	,	,	PUNCT
ejpam-4396	65	27	but	but	CCONJ
ejpam-4396	65	28	still	still	ADV
ejpam-4396	65	29	performs	perform	VERB
ejpam-4396	65	30	the	the	DET
ejpam-4396	65	31	same	same	ADJ
ejpam-4396	65	32	function	function	NOUN
ejpam-4396	65	33	as	as	ADP
ejpam-4396	65	34	(	(	PUNCT
ejpam-4396	65	35	a	a	DET
ejpam-4396	65	36	∨	∨	NUM
ejpam-4396	65	37	b	b	NOUN
ejpam-4396	65	38	)	)	PUNCT
ejpam-4396	65	39	∨	∨	NOUN
ejpam-4396	65	40	(	(	PUNCT
ejpam-4396	65	41	a	a	DET
ejpam-4396	65	42	∨	∨	NUM
ejpam-4396	65	43	c	c	NOUN
ejpam-4396	65	44	)	)	PUNCT
ejpam-4396	65	45	.	.	PUNCT
ejpam-4396	66	1	this	this	PRON
ejpam-4396	66	2	simplifies	simplify	VERB
ejpam-4396	66	3	the	the	DET
ejpam-4396	66	4	design	design	NOUN
ejpam-4396	66	5	of	of	ADP
ejpam-4396	66	6	the	the	DET
ejpam-4396	66	7	circuit	circuit	NOUN
ejpam-4396	66	8	.	.	PUNCT
ejpam-4396	67	1	in	in	ADP
ejpam-4396	67	2	this	this	DET
ejpam-4396	67	3	study	study	NOUN
ejpam-4396	67	4	,	,	PUNCT
ejpam-4396	67	5	we	we	PRON
ejpam-4396	67	6	gave	give	VERB
ejpam-4396	67	7	some	some	DET
ejpam-4396	67	8	important	important	ADJ
ejpam-4396	67	9	properties	property	NOUN
ejpam-4396	67	10	of	of	ADP
ejpam-4396	67	11	g	g	NOUN
ejpam-4396	67	12	-	-	PUNCT
ejpam-4396	67	13	groups	group	NOUN
ejpam-4396	67	14	,	,	PUNCT
ejpam-4396	67	15	and	and	CCONJ
ejpam-4396	67	16	provided	provide	VERB
ejpam-4396	67	17	a	a	DET
ejpam-4396	67	18	couple	couple	NOUN
ejpam-4396	67	19	of	of	ADP
ejpam-4396	67	20	ways	way	NOUN
ejpam-4396	67	21	of	of	ADP
ejpam-4396	67	22	constructing	construct	VERB
ejpam-4396	67	23	them	they	PRON
ejpam-4396	67	24	.	.	PUNCT
ejpam-4396	68	1	2	2	X
ejpam-4396	68	2	.	.	X
ejpam-4396	68	3	preliminary	preliminary	ADJ
ejpam-4396	68	4	results	result	NOUN
ejpam-4396	68	5	the	the	DET
ejpam-4396	68	6	following	following	ADJ
ejpam-4396	68	7	statements	statement	NOUN
ejpam-4396	68	8	are	be	AUX
ejpam-4396	68	9	found	find	VERB
ejpam-4396	68	10	in	in	ADP
ejpam-4396	68	11	[	[	X
ejpam-4396	68	12	1	1	NUM
ejpam-4396	68	13	]	]	PUNCT
ejpam-4396	68	14	.	.	PUNCT
ejpam-4396	69	1	we	we	PRON
ejpam-4396	69	2	shall	shall	AUX
ejpam-4396	69	3	be	be	AUX
ejpam-4396	69	4	using	use	VERB
ejpam-4396	69	5	them	they	PRON
ejpam-4396	69	6	for	for	ADP
ejpam-4396	69	7	the	the	DET
ejpam-4396	69	8	succeeding	succeed	VERB
ejpam-4396	69	9	properties	property	NOUN
ejpam-4396	69	10	.	.	PUNCT
ejpam-4396	70	1	remark	remark	PROPN
ejpam-4396	70	2	1	1	NUM
ejpam-4396	70	3	.	.	PUNCT
ejpam-4396	71	1	an	an	DET
ejpam-4396	71	2	inverse	inverse	NOUN
ejpam-4396	71	3	of	of	ADP
ejpam-4396	71	4	a	a	DET
ejpam-4396	71	5	unit	unit	NOUN
ejpam-4396	71	6	is	be	AUX
ejpam-4396	71	7	also	also	ADV
ejpam-4396	71	8	a	a	DET
ejpam-4396	71	9	unit	unit	NOUN
ejpam-4396	71	10	.	.	PUNCT
ejpam-4396	72	1	in	in	ADP
ejpam-4396	72	2	addition	addition	NOUN
ejpam-4396	72	3	,	,	PUNCT
ejpam-4396	72	4	the	the	DET
ejpam-4396	72	5	two	two	NUM
ejpam-4396	72	6	(	(	PUNCT
ejpam-4396	72	7	the	the	DET
ejpam-4396	72	8	unit	unit	NOUN
ejpam-4396	72	9	and	and	CCONJ
ejpam-4396	72	10	its	its	PRON
ejpam-4396	72	11	inverse	inverse	NOUN
ejpam-4396	72	12	)	)	PUNCT
ejpam-4396	72	13	have	have	VERB
ejpam-4396	72	14	the	the	DET
ejpam-4396	72	15	same	same	ADJ
ejpam-4396	72	16	identity	identity	NOUN
ejpam-4396	72	17	element	element	NOUN
ejpam-4396	72	18	.	.	PUNCT
ejpam-4396	73	1	remark	remark	NOUN
ejpam-4396	73	2	2	2	NUM
ejpam-4396	73	3	.	.	PUNCT
ejpam-4396	74	1	a	a	DET
ejpam-4396	74	2	unit	unit	NOUN
ejpam-4396	74	3	has	have	VERB
ejpam-4396	74	4	a	a	DET
ejpam-4396	74	5	unique	unique	ADJ
ejpam-4396	74	6	inverse	inverse	NOUN
ejpam-4396	74	7	.	.	PUNCT
ejpam-4396	75	1	remark	remark	NOUN
ejpam-4396	75	2	3	3	NUM
ejpam-4396	75	3	.	.	PUNCT
ejpam-4396	76	1	the	the	DET
ejpam-4396	76	2	identity	identity	NOUN
ejpam-4396	76	3	of	of	ADP
ejpam-4396	76	4	a	a	DET
ejpam-4396	76	5	unit	unit	NOUN
ejpam-4396	76	6	is	be	AUX
ejpam-4396	76	7	also	also	ADV
ejpam-4396	76	8	a	a	DET
ejpam-4396	76	9	unit	unit	NOUN
ejpam-4396	76	10	.	.	PUNCT
ejpam-4396	77	1	remark	remark	PROPN
ejpam-4396	77	2	4	4	NUM
ejpam-4396	77	3	.	.	PUNCT
ejpam-4396	78	1	in	in	ADP
ejpam-4396	78	2	an	an	DET
ejpam-4396	78	3	abelian	abelian	ADJ
ejpam-4396	78	4	g	g	PROPN
ejpam-4396	78	5	-	-	PUNCT
ejpam-4396	78	6	group	group	NOUN
ejpam-4396	78	7	,	,	PUNCT
ejpam-4396	78	8	the	the	DET
ejpam-4396	78	9	identity	identity	NOUN
ejpam-4396	78	10	of	of	ADP
ejpam-4396	78	11	the	the	DET
ejpam-4396	78	12	product	product	NOUN
ejpam-4396	78	13	of	of	ADP
ejpam-4396	78	14	two	two	NUM
ejpam-4396	78	15	units	unit	NOUN
ejpam-4396	78	16	is	be	AUX
ejpam-4396	78	17	equal	equal	ADJ
ejpam-4396	78	18	to	to	ADP
ejpam-4396	78	19	the	the	DET
ejpam-4396	78	20	product	product	NOUN
ejpam-4396	78	21	of	of	ADP
ejpam-4396	78	22	their	their	PRON
ejpam-4396	78	23	corresponding	correspond	VERB
ejpam-4396	78	24	identities	identity	NOUN
ejpam-4396	78	25	.	.	PUNCT
ejpam-4396	79	1	j.	j.	PROPN
ejpam-4396	79	2	caraquil	caraquil	PROPN
ejpam-4396	79	3	,	,	PUNCT
ejpam-4396	79	4	m.	m.	PROPN
ejpam-4396	79	5	baldado	baldado	PROPN
ejpam-4396	79	6	jr	jr	PROPN
ejpam-4396	79	7	.	.	PROPN
ejpam-4396	79	8	/	/	SYM
ejpam-4396	79	9	eur	eur	PROPN
ejpam-4396	79	10	.	.	PUNCT
ejpam-4396	80	1	j.	j.	PROPN
ejpam-4396	80	2	pure	pure	PROPN
ejpam-4396	80	3	appl	appl	PROPN
ejpam-4396	80	4	.	.	PROPN
ejpam-4396	80	5	math	math	PROPN
ejpam-4396	80	6	,	,	PUNCT
ejpam-4396	80	7	15	15	NUM
ejpam-4396	80	8	(	(	PUNCT
ejpam-4396	80	9	3	3	NUM
ejpam-4396	80	10	)	)	PUNCT
ejpam-4396	80	11	(	(	PUNCT
ejpam-4396	80	12	2022	2022	NUM
ejpam-4396	80	13	)	)	PUNCT
ejpam-4396	80	14	,	,	PUNCT
ejpam-4396	80	15	887	887	NUM
ejpam-4396	80	16	-	-	SYM
ejpam-4396	80	17	896	896	NUM
ejpam-4396	80	18	890	890	NUM
ejpam-4396	80	19	let	let	VERB
ejpam-4396	80	20	g	g	NOUN
ejpam-4396	80	21	be	be	AUX
ejpam-4396	80	22	an	an	DET
ejpam-4396	80	23	abelian	abelian	ADJ
ejpam-4396	80	24	g	g	NOUN
ejpam-4396	80	25	-	-	PUNCT
ejpam-4396	80	26	group	group	NOUN
ejpam-4396	80	27	,	,	PUNCT
ejpam-4396	80	28	and	and	CCONJ
ejpam-4396	80	29	h	h	NOUN
ejpam-4396	80	30	be	be	VERB
ejpam-4396	80	31	the	the	DET
ejpam-4396	80	32	set	set	NOUN
ejpam-4396	80	33	of	of	ADP
ejpam-4396	80	34	all	all	DET
ejpam-4396	80	35	units	unit	NOUN
ejpam-4396	80	36	of	of	ADP
ejpam-4396	80	37	g	g	NOUN
ejpam-4396	80	38	,	,	PUNCT
ejpam-4396	80	39	that	that	PRON
ejpam-4396	80	40	is	be	AUX
ejpam-4396	80	41	h	h	NOUN
ejpam-4396	80	42	=	=	PUNCT
ejpam-4396	80	43	{	{	PUNCT
ejpam-4396	80	44	h	h	NOUN
ejpam-4396	80	45	∈	∈	PROPN
ejpam-4396	80	46	g	g	PROPN
ejpam-4396	80	47	:	:	PUNCT
ejpam-4396	80	48	h	h	PROPN
ejpam-4396	80	49	is	be	AUX
ejpam-4396	80	50	a	a	DET
ejpam-4396	80	51	unit	unit	NOUN
ejpam-4396	80	52	}	}	PUNCT
ejpam-4396	80	53	.	.	PUNCT
ejpam-4396	81	1	in	in	ADP
ejpam-4396	81	2	the	the	DET
ejpam-4396	81	3	succeeding	succeed	VERB
ejpam-4396	81	4	discussions	discussion	NOUN
ejpam-4396	81	5	,	,	PUNCT
ejpam-4396	81	6	the	the	DET
ejpam-4396	81	7	set	set	NOUN
ejpam-4396	81	8	h	h	NOUN
ejpam-4396	81	9	refer	refer	VERB
ejpam-4396	81	10	to	to	ADP
ejpam-4396	81	11	h	h	NOUN
ejpam-4396	81	12	=	=	PRON
ejpam-4396	81	13	{	{	PUNCT
ejpam-4396	81	14	h	h	NOUN
ejpam-4396	81	15	∈	∈	PROPN
ejpam-4396	81	16	g	g	PROPN
ejpam-4396	81	17	:	:	PUNCT
ejpam-4396	81	18	h	h	PROPN
ejpam-4396	81	19	is	be	AUX
ejpam-4396	81	20	a	a	DET
ejpam-4396	81	21	unit	unit	NOUN
ejpam-4396	81	22	}	}	PUNCT
ejpam-4396	81	23	.	.	PUNCT
ejpam-4396	82	1	we	we	PRON
ejpam-4396	82	2	say	say	VERB
ejpam-4396	82	3	that	that	SCONJ
ejpam-4396	82	4	h	h	NOUN
ejpam-4396	82	5	has	have	VERB
ejpam-4396	82	6	a	a	DET
ejpam-4396	82	7	unique	unique	ADJ
ejpam-4396	82	8	identity	identity	NOUN
ejpam-4396	82	9	(	(	PUNCT
ejpam-4396	82	10	or	or	CCONJ
ejpam-4396	82	11	a	a	DET
ejpam-4396	82	12	trunk	trunk	NOUN
ejpam-4396	82	13	)	)	PUNCT
ejpam-4396	82	14	if	if	SCONJ
ejpam-4396	82	15	all	all	DET
ejpam-4396	82	16	the	the	DET
ejpam-4396	82	17	elements	element	NOUN
ejpam-4396	82	18	of	of	ADP
ejpam-4396	82	19	h	h	NOUN
ejpam-4396	82	20	have	have	VERB
ejpam-4396	82	21	the	the	DET
ejpam-4396	82	22	same	same	ADJ
ejpam-4396	82	23	identity	identity	NOUN
ejpam-4396	82	24	element	element	NOUN
ejpam-4396	82	25	.	.	PUNCT
ejpam-4396	83	1	remark	remark	NOUN
ejpam-4396	83	2	5	5	NUM
ejpam-4396	83	3	.	.	PUNCT
ejpam-4396	84	1	if	if	SCONJ
ejpam-4396	84	2	x	x	SYM
ejpam-4396	84	3	∈	∈	PROPN
ejpam-4396	84	4	g\h	g\h	PROPN
ejpam-4396	84	5	,	,	PUNCT
ejpam-4396	84	6	then	then	ADV
ejpam-4396	84	7	x	x	PUNCT
ejpam-4396	84	8	has	have	VERB
ejpam-4396	84	9	a	a	DET
ejpam-4396	84	10	unique	unique	ADJ
ejpam-4396	84	11	identity	identity	NOUN
ejpam-4396	84	12	element	element	NOUN
ejpam-4396	84	13	for	for	ADP
ejpam-4396	84	14	which	which	PRON
ejpam-4396	84	15	it	it	PRON
ejpam-4396	84	16	has	have	VERB
ejpam-4396	84	17	an	an	DET
ejpam-4396	84	18	inverse	inverse	NOUN
ejpam-4396	84	19	.	.	PUNCT
ejpam-4396	85	1	3	3	X
ejpam-4396	85	2	.	.	X
ejpam-4396	85	3	g	g	NOUN
ejpam-4396	85	4	-	-	PUNCT
ejpam-4396	85	5	subgroups	subgroup	NOUN
ejpam-4396	85	6	in	in	ADP
ejpam-4396	85	7	this	this	DET
ejpam-4396	85	8	section	section	NOUN
ejpam-4396	85	9	,	,	PUNCT
ejpam-4396	85	10	we	we	PRON
ejpam-4396	85	11	introduce	introduce	VERB
ejpam-4396	85	12	the	the	DET
ejpam-4396	85	13	concept	concept	NOUN
ejpam-4396	85	14	g	g	NOUN
ejpam-4396	85	15	-	-	PUNCT
ejpam-4396	85	16	subgroups	subgroup	NOUN
ejpam-4396	85	17	,	,	PUNCT
ejpam-4396	85	18	and	and	CCONJ
ejpam-4396	85	19	provide	provide	VERB
ejpam-4396	85	20	some	some	DET
ejpam-4396	85	21	means	mean	NOUN
ejpam-4396	85	22	of	of	ADP
ejpam-4396	85	23	constructing	construct	VERB
ejpam-4396	85	24	them	they	PRON
ejpam-4396	85	25	.	.	PUNCT
ejpam-4396	86	1	a	a	DET
ejpam-4396	86	2	very	very	ADV
ejpam-4396	86	3	distinctive	distinctive	ADJ
ejpam-4396	86	4	property	property	NOUN
ejpam-4396	86	5	of	of	ADP
ejpam-4396	86	6	some	some	DET
ejpam-4396	86	7	types	type	NOUN
ejpam-4396	86	8	of	of	ADP
ejpam-4396	86	9	g	g	NOUN
ejpam-4396	86	10	-	-	PUNCT
ejpam-4396	86	11	subgroups	subgroup	NOUN
ejpam-4396	86	12	is	be	AUX
ejpam-4396	86	13	that	that	SCONJ
ejpam-4396	86	14	their	their	PRON
ejpam-4396	86	15	complements	complement	NOUN
ejpam-4396	86	16	are	be	AUX
ejpam-4396	86	17	also	also	ADV
ejpam-4396	86	18	g	g	NOUN
ejpam-4396	86	19	-	-	PUNCT
ejpam-4396	86	20	subgroups	subgroup	NOUN
ejpam-4396	86	21	,	,	PUNCT
ejpam-4396	86	22	which	which	PRON
ejpam-4396	86	23	is	be	AUX
ejpam-4396	86	24	not	not	PART
ejpam-4396	86	25	always	always	ADV
ejpam-4396	86	26	the	the	DET
ejpam-4396	86	27	case	case	NOUN
ejpam-4396	86	28	in	in	ADP
ejpam-4396	86	29	other	other	ADJ
ejpam-4396	86	30	structures	structure	NOUN
ejpam-4396	86	31	.	.	PUNCT
ejpam-4396	87	1	most	most	ADJ
ejpam-4396	87	2	of	of	ADP
ejpam-4396	87	3	the	the	DET
ejpam-4396	87	4	discussions	discussion	NOUN
ejpam-4396	87	5	in	in	ADP
ejpam-4396	87	6	this	this	DET
ejpam-4396	87	7	section	section	NOUN
ejpam-4396	87	8	is	be	AUX
ejpam-4396	87	9	focused	focus	VERB
ejpam-4396	87	10	on	on	ADP
ejpam-4396	87	11	showing	show	VERB
ejpam-4396	87	12	that	that	SCONJ
ejpam-4396	87	13	the	the	DET
ejpam-4396	87	14	set	set	NOUN
ejpam-4396	87	15	of	of	ADP
ejpam-4396	87	16	all	all	DET
ejpam-4396	87	17	units	unit	NOUN
ejpam-4396	87	18	of	of	ADP
ejpam-4396	87	19	a	a	DET
ejpam-4396	87	20	particular	particular	ADJ
ejpam-4396	87	21	g	g	NOUN
ejpam-4396	87	22	-	-	PUNCT
ejpam-4396	87	23	group	group	NOUN
ejpam-4396	87	24	possesses	possess	VERB
ejpam-4396	87	25	the	the	DET
ejpam-4396	87	26	said	say	VERB
ejpam-4396	87	27	property	property	NOUN
ejpam-4396	87	28	.	.	PUNCT
ejpam-4396	88	1	the	the	DET
ejpam-4396	88	2	next	next	ADJ
ejpam-4396	88	3	statement	statement	NOUN
ejpam-4396	88	4	,	,	PUNCT
ejpam-4396	88	5	definition	definition	NOUN
ejpam-4396	88	6	1	1	NUM
ejpam-4396	88	7	,	,	PUNCT
ejpam-4396	88	8	defines	define	VERB
ejpam-4396	88	9	what	what	PRON
ejpam-4396	88	10	a	a	DET
ejpam-4396	88	11	g	g	NOUN
ejpam-4396	88	12	-	-	PUNCT
ejpam-4396	88	13	subgroup	subgroup	NOUN
ejpam-4396	88	14	is	be	AUX
ejpam-4396	88	15	.	.	PUNCT
ejpam-4396	89	1	definition	definition	NOUN
ejpam-4396	89	2	1	1	NUM
ejpam-4396	89	3	.	.	PUNCT
ejpam-4396	90	1	let	let	VERB
ejpam-4396	90	2	g	g	PRON
ejpam-4396	90	3	be	be	AUX
ejpam-4396	90	4	a	a	DET
ejpam-4396	90	5	g	g	NOUN
ejpam-4396	90	6	-	-	PUNCT
ejpam-4396	90	7	group	group	NOUN
ejpam-4396	90	8	.	.	PUNCT
ejpam-4396	91	1	a	a	DET
ejpam-4396	91	2	non	non	ADJ
ejpam-4396	91	3	-	-	ADJ
ejpam-4396	91	4	empty	empty	ADJ
ejpam-4396	91	5	subset	subset	NOUN
ejpam-4396	91	6	j	j	PROPN
ejpam-4396	91	7	of	of	ADP
ejpam-4396	91	8	g	g	PROPN
ejpam-4396	91	9	is	be	AUX
ejpam-4396	91	10	called	call	VERB
ejpam-4396	91	11	a	a	DET
ejpam-4396	91	12	g	g	NOUN
ejpam-4396	91	13	-	-	PUNCT
ejpam-4396	91	14	subgroup	subgroup	NOUN
ejpam-4396	91	15	of	of	ADP
ejpam-4396	91	16	g	g	PROPN
ejpam-4396	91	17	if	if	SCONJ
ejpam-4396	91	18	j	j	PROPN
ejpam-4396	91	19	is	be	AUX
ejpam-4396	91	20	a	a	DET
ejpam-4396	91	21	g	g	NOUN
ejpam-4396	91	22	-	-	PUNCT
ejpam-4396	91	23	group	group	NOUN
ejpam-4396	91	24	with	with	ADP
ejpam-4396	91	25	respect	respect	NOUN
ejpam-4396	91	26	to	to	ADP
ejpam-4396	91	27	the	the	DET
ejpam-4396	91	28	operation	operation	NOUN
ejpam-4396	91	29	of	of	ADP
ejpam-4396	91	30	g.	g.	PROPN
ejpam-4396	91	31	example	example	PROPN
ejpam-4396	91	32	1	1	NUM
ejpam-4396	91	33	exhibits	exhibit	VERB
ejpam-4396	91	34	some	some	PRON
ejpam-4396	91	35	of	of	ADP
ejpam-4396	91	36	g	g	NOUN
ejpam-4396	91	37	-	-	PUNCT
ejpam-4396	91	38	subgroups	subgroup	NOUN
ejpam-4396	91	39	.	.	PUNCT
ejpam-4396	92	1	theorem	theorem	NOUN
ejpam-4396	92	2	1	1	NUM
ejpam-4396	92	3	.	.	PUNCT
ejpam-4396	93	1	let	let	VERB
ejpam-4396	93	2	g	g	NOUN
ejpam-4396	93	3	be	be	AUX
ejpam-4396	93	4	an	an	DET
ejpam-4396	93	5	abelian	abelian	ADJ
ejpam-4396	93	6	g	g	NOUN
ejpam-4396	93	7	-	-	PUNCT
ejpam-4396	93	8	group	group	NOUN
ejpam-4396	93	9	.	.	PUNCT
ejpam-4396	94	1	if	if	SCONJ
ejpam-4396	94	2	h	h	NOUN
ejpam-4396	94	3	̸=	̸=	PROPN
ejpam-4396	94	4	∅	∅	NOUN
ejpam-4396	94	5	,	,	PUNCT
ejpam-4396	94	6	then	then	ADV
ejpam-4396	94	7	h	h	PROPN
ejpam-4396	94	8	is	be	AUX
ejpam-4396	94	9	a	a	DET
ejpam-4396	94	10	g	g	NOUN
ejpam-4396	94	11	-	-	PUNCT
ejpam-4396	94	12	subgroup	subgroup	NOUN
ejpam-4396	94	13	of	of	ADP
ejpam-4396	94	14	g.	g.	PROPN
ejpam-4396	94	15	proof	proof	PROPN
ejpam-4396	94	16	.	.	PUNCT
ejpam-4396	95	1	(	(	PUNCT
ejpam-4396	95	2	g1	g1	PROPN
ejpam-4396	95	3	)	)	PUNCT
ejpam-4396	95	4	is	be	AUX
ejpam-4396	95	5	satisfied	satisfied	ADJ
ejpam-4396	95	6	by	by	ADP
ejpam-4396	95	7	the	the	DET
ejpam-4396	95	8	fact	fact	NOUN
ejpam-4396	95	9	that	that	SCONJ
ejpam-4396	95	10	h	h	NOUN
ejpam-4396	95	11	is	be	AUX
ejpam-4396	95	12	a	a	DET
ejpam-4396	95	13	subset	subset	NOUN
ejpam-4396	95	14	of	of	ADP
ejpam-4396	95	15	g.	g.	PROPN
ejpam-4396	95	16	(	(	PUNCT
ejpam-4396	95	17	g2	g2	PROPN
ejpam-4396	95	18	)	)	PUNCT
ejpam-4396	95	19	follows	follow	VERB
ejpam-4396	95	20	from	from	ADP
ejpam-4396	95	21	remark	remark	NOUN
ejpam-4396	95	22	3	3	NUM
ejpam-4396	95	23	,	,	PUNCT
ejpam-4396	95	24	and	and	CCONJ
ejpam-4396	95	25	(	(	PUNCT
ejpam-4396	95	26	g3	g3	NOUN
ejpam-4396	95	27	)	)	PUNCT
ejpam-4396	95	28	follows	follow	VERB
ejpam-4396	95	29	from	from	ADP
ejpam-4396	95	30	remark	remark	NOUN
ejpam-4396	95	31	1	1	NUM
ejpam-4396	95	32	and	and	CCONJ
ejpam-4396	95	33	remark	remark	NOUN
ejpam-4396	95	34	2	2	NUM
ejpam-4396	95	35	.	.	PUNCT
ejpam-4396	96	1	what	what	PRON
ejpam-4396	96	2	remains	remain	VERB
ejpam-4396	96	3	to	to	PART
ejpam-4396	96	4	be	be	AUX
ejpam-4396	96	5	shown	show	VERB
ejpam-4396	96	6	is	be	AUX
ejpam-4396	96	7	the	the	DET
ejpam-4396	96	8	fact	fact	NOUN
ejpam-4396	96	9	that	that	SCONJ
ejpam-4396	96	10	the	the	DET
ejpam-4396	96	11	operation	operation	NOUN
ejpam-4396	96	12	is	be	AUX
ejpam-4396	96	13	a	a	DET
ejpam-4396	96	14	binary	binary	ADJ
ejpam-4396	96	15	operation	operation	NOUN
ejpam-4396	96	16	in	in	ADP
ejpam-4396	96	17	h	h	NOUN
ejpam-4396	96	18	,	,	PUNCT
ejpam-4396	96	19	that	that	ADV
ejpam-4396	96	20	is	is	ADV
ejpam-4396	96	21	,	,	PUNCT
ejpam-4396	96	22	the	the	DET
ejpam-4396	96	23	product	product	NOUN
ejpam-4396	96	24	of	of	ADP
ejpam-4396	96	25	two	two	NUM
ejpam-4396	96	26	units	unit	NOUN
ejpam-4396	96	27	is	be	AUX
ejpam-4396	96	28	itself	itself	PRON
ejpam-4396	96	29	a	a	DET
ejpam-4396	96	30	unit	unit	NOUN
ejpam-4396	96	31	.	.	PUNCT
ejpam-4396	97	1	let	let	VERB
ejpam-4396	97	2	a	a	DET
ejpam-4396	97	3	,	,	PUNCT
ejpam-4396	97	4	b	b	PROPN
ejpam-4396	97	5	∈	∈	PROPN
ejpam-4396	97	6	h.	h.	NOUN
ejpam-4396	97	7	if	if	SCONJ
ejpam-4396	97	8	e′	e′	PROPN
ejpam-4396	97	9	is	be	AUX
ejpam-4396	97	10	an	an	DET
ejpam-4396	97	11	identity	identity	NOUN
ejpam-4396	97	12	of	of	ADP
ejpam-4396	97	13	ab	ab	PROPN
ejpam-4396	97	14	,	,	PUNCT
ejpam-4396	97	15	then	then	ADV
ejpam-4396	97	16	by	by	ADP
ejpam-4396	97	17	remark	remark	NOUN
ejpam-4396	97	18	4	4	NUM
ejpam-4396	97	19	,	,	PUNCT
ejpam-4396	97	20	e′	e′	X
ejpam-4396	97	21	=	=	SYM
ejpam-4396	97	22	eaeb	eaeb	PROPN
ejpam-4396	97	23	=	=	SYM
ejpam-4396	97	24	eab	eab	PROPN
ejpam-4396	97	25	.	.	PUNCT
ejpam-4396	98	1	this	this	PRON
ejpam-4396	98	2	shows	show	VERB
ejpam-4396	98	3	that	that	SCONJ
ejpam-4396	98	4	ab	ab	PROPN
ejpam-4396	98	5	has	have	VERB
ejpam-4396	98	6	a	a	DET
ejpam-4396	98	7	unique	unique	ADJ
ejpam-4396	98	8	identity	identity	NOUN
ejpam-4396	98	9	,	,	PUNCT
ejpam-4396	98	10	that	that	ADV
ejpam-4396	98	11	is	is	ADV
ejpam-4396	98	12	,	,	PUNCT
ejpam-4396	98	13	ab	ab	PROPN
ejpam-4396	98	14	is	be	AUX
ejpam-4396	98	15	a	a	DET
ejpam-4396	98	16	unit	unit	NOUN
ejpam-4396	98	17	.	.	PUNCT
ejpam-4396	99	1	the	the	DET
ejpam-4396	99	2	next	next	ADJ
ejpam-4396	99	3	statement	statement	NOUN
ejpam-4396	99	4	,	,	PUNCT
ejpam-4396	99	5	lemma	lemma	PROPN
ejpam-4396	99	6	1	1	NUM
ejpam-4396	99	7	,	,	PUNCT
ejpam-4396	99	8	says	say	VERB
ejpam-4396	99	9	that	that	SCONJ
ejpam-4396	99	10	in	in	ADP
ejpam-4396	99	11	an	an	DET
ejpam-4396	99	12	abelian	abelian	ADJ
ejpam-4396	99	13	group	group	NOUN
ejpam-4396	99	14	every	every	DET
ejpam-4396	99	15	element	element	NOUN
ejpam-4396	99	16	of	of	ADP
ejpam-4396	99	17	h	h	NOUN
ejpam-4396	99	18	has	have	VERB
ejpam-4396	99	19	the	the	DET
ejpam-4396	99	20	same	same	ADJ
ejpam-4396	99	21	identity	identity	NOUN
ejpam-4396	99	22	element	element	NOUN
ejpam-4396	99	23	,	,	PUNCT
ejpam-4396	99	24	that	that	PRON
ejpam-4396	99	25	is	is	ADV
ejpam-4396	99	26	h	h	NOUN
ejpam-4396	99	27	is	be	AUX
ejpam-4396	99	28	a	a	DET
ejpam-4396	99	29	trunk	trunk	NOUN
ejpam-4396	99	30	.	.	PUNCT
ejpam-4396	100	1	lemma	lemma	PROPN
ejpam-4396	100	2	1	1	X
ejpam-4396	100	3	.	.	PUNCT
ejpam-4396	101	1	let	let	VERB
ejpam-4396	101	2	g	g	NOUN
ejpam-4396	101	3	be	be	AUX
ejpam-4396	101	4	an	an	DET
ejpam-4396	101	5	abelian	abelian	ADJ
ejpam-4396	101	6	g	g	NOUN
ejpam-4396	101	7	-	-	PUNCT
ejpam-4396	101	8	group	group	NOUN
ejpam-4396	101	9	.	.	PUNCT
ejpam-4396	102	1	if	if	SCONJ
ejpam-4396	102	2	a	a	DET
ejpam-4396	102	3	,	,	PUNCT
ejpam-4396	102	4	b	b	X
ejpam-4396	102	5	∈	∈	PROPN
ejpam-4396	102	6	h	h	NOUN
ejpam-4396	102	7	,	,	PUNCT
ejpam-4396	102	8	then	then	ADV
ejpam-4396	102	9	ea	ea	PROPN
ejpam-4396	102	10	=	=	SYM
ejpam-4396	102	11	eb	eb	PROPN
ejpam-4396	102	12	.	.	PUNCT
ejpam-4396	102	13	proof	proof	NOUN
ejpam-4396	102	14	.	.	PUNCT
ejpam-4396	103	1	let	let	VERB
ejpam-4396	103	2	a	a	DET
ejpam-4396	103	3	,	,	PUNCT
ejpam-4396	103	4	b	b	PROPN
ejpam-4396	103	5	∈	∈	PROPN
ejpam-4396	103	6	h.	h.	NOUN
ejpam-4396	103	7	then	then	ADV
ejpam-4396	103	8	by	by	ADP
ejpam-4396	103	9	theorem	theorem	NOUN
ejpam-4396	103	10	1	1	NUM
ejpam-4396	103	11	,	,	PUNCT
ejpam-4396	103	12	ab	ab	PROPN
ejpam-4396	103	13	is	be	AUX
ejpam-4396	103	14	a	a	DET
ejpam-4396	103	15	unit	unit	NOUN
ejpam-4396	103	16	.	.	PUNCT
ejpam-4396	104	1	observe	observe	VERB
ejpam-4396	104	2	that	that	DET
ejpam-4396	104	3	eaab	eaab	PROPN
ejpam-4396	104	4	=	=	SYM
ejpam-4396	104	5	ab	ab	PROPN
ejpam-4396	104	6	,	,	PUNCT
ejpam-4396	104	7	whence	whence	NOUN
ejpam-4396	104	8	,	,	PUNCT
ejpam-4396	104	9	ea	ea	NOUN
ejpam-4396	105	1	=	=	PUNCT
ejpam-4396	105	2	eab	eab	PROPN
ejpam-4396	105	3	.	.	PUNCT
ejpam-4396	106	1	similarly	similarly	ADV
ejpam-4396	106	2	,	,	PUNCT
ejpam-4396	106	3	observe	observe	VERB
ejpam-4396	106	4	that	that	SCONJ
ejpam-4396	106	5	abeb	abeb	NOUN
ejpam-4396	106	6	=	=	SYM
ejpam-4396	106	7	ab	ab	PROPN
ejpam-4396	106	8	,	,	PUNCT
ejpam-4396	106	9	whence	whence	PROPN
ejpam-4396	106	10	,	,	PUNCT
ejpam-4396	106	11	eb	eb	PROPN
ejpam-4396	106	12	=	=	SYM
ejpam-4396	106	13	eab	eab	PROPN
ejpam-4396	106	14	.	.	PUNCT
ejpam-4396	107	1	since	since	SCONJ
ejpam-4396	107	2	a	a	DET
ejpam-4396	107	3	,	,	PUNCT
ejpam-4396	107	4	b	b	NOUN
ejpam-4396	107	5	,	,	PUNCT
ejpam-4396	107	6	and	and	CCONJ
ejpam-4396	107	7	ab	ab	PROPN
ejpam-4396	107	8	are	be	AUX
ejpam-4396	107	9	units	unit	NOUN
ejpam-4396	107	10	,	,	PUNCT
ejpam-4396	107	11	we	we	PRON
ejpam-4396	107	12	have	have	VERB
ejpam-4396	107	13	ea	ea	NOUN
ejpam-4396	107	14	=	=	SYM
ejpam-4396	107	15	eab	eab	NOUN
ejpam-4396	107	16	=	=	SYM
ejpam-4396	107	17	eb	eb	PROPN
ejpam-4396	107	18	.	.	PUNCT
ejpam-4396	108	1	the	the	DET
ejpam-4396	108	2	next	next	ADJ
ejpam-4396	108	3	statement	statement	NOUN
ejpam-4396	108	4	follows	follow	VERB
ejpam-4396	108	5	from	from	ADP
ejpam-4396	108	6	theorem	theorem	ADJ
ejpam-4396	108	7	1	1	NUM
ejpam-4396	108	8	and	and	CCONJ
ejpam-4396	108	9	lemma	lemma	PROPN
ejpam-4396	108	10	1	1	PROPN
ejpam-4396	108	11	.	.	PUNCT
ejpam-4396	108	12	recall	recall	PROPN
ejpam-4396	108	13	,	,	PUNCT
ejpam-4396	108	14	a	a	DET
ejpam-4396	108	15	group	group	NOUN
ejpam-4396	108	16	is	be	AUX
ejpam-4396	108	17	a	a	DET
ejpam-4396	108	18	non	non	ADJ
ejpam-4396	108	19	-	-	ADJ
ejpam-4396	108	20	empty	empty	ADJ
ejpam-4396	108	21	set	set	NOUN
ejpam-4396	108	22	g	g	NOUN
ejpam-4396	108	23	together	together	ADV
ejpam-4396	108	24	with	with	ADP
ejpam-4396	108	25	a	a	DET
ejpam-4396	108	26	binary	binary	ADJ
ejpam-4396	108	27	operation	operation	NOUN
ejpam-4396	108	28	∗	∗	NOUN
ejpam-4396	108	29	,	,	PUNCT
ejpam-4396	108	30	satisfying	satisfy	VERB
ejpam-4396	108	31	the	the	DET
ejpam-4396	108	32	following	follow	VERB
ejpam-4396	108	33	axioms	axiom	NOUN
ejpam-4396	108	34	:	:	PUNCT
ejpam-4396	108	35	(	(	PUNCT
ejpam-4396	108	36	g1	g1	PROPN
ejpam-4396	108	37	)	)	PUNCT
ejpam-4396	108	38	for	for	ADP
ejpam-4396	108	39	all	all	DET
ejpam-4396	108	40	a	a	DET
ejpam-4396	108	41	,	,	PUNCT
ejpam-4396	108	42	b	b	NOUN
ejpam-4396	108	43	,	,	PUNCT
ejpam-4396	108	44	c	c	PROPN
ejpam-4396	108	45	∈	∈	PROPN
ejpam-4396	108	46	g	g	PROPN
ejpam-4396	108	47	,	,	PUNCT
ejpam-4396	108	48	we	we	PRON
ejpam-4396	108	49	have	have	VERB
ejpam-4396	108	50	(	(	PUNCT
ejpam-4396	108	51	a	a	DET
ejpam-4396	108	52	∗	∗	NOUN
ejpam-4396	108	53	b	b	NOUN
ejpam-4396	108	54	)	)	PUNCT
ejpam-4396	108	55	∗	∗	NOUN
ejpam-4396	108	56	c	c	NOUN
ejpam-4396	108	57	=	=	PUNCT
ejpam-4396	108	58	a	a	DET
ejpam-4396	108	59	∗	∗	NOUN
ejpam-4396	108	60	(	(	PUNCT
ejpam-4396	108	61	b	b	NOUN
ejpam-4396	108	62	∗	∗	NOUN
ejpam-4396	108	63	c	c	NOUN
ejpam-4396	108	64	)	)	PUNCT
ejpam-4396	108	65	;	;	PUNCT
ejpam-4396	108	66	(	(	PUNCT
ejpam-4396	108	67	g2	g2	PROPN
ejpam-4396	108	68	)	)	PUNCT
ejpam-4396	108	69	there	there	PRON
ejpam-4396	108	70	is	be	VERB
ejpam-4396	108	71	an	an	DET
ejpam-4396	108	72	element	element	NOUN
ejpam-4396	108	73	e	e	NOUN
ejpam-4396	108	74	in	in	ADP
ejpam-4396	108	75	g	g	PROPN
ejpam-4396	108	76	such	such	ADJ
ejpam-4396	108	77	that	that	PRON
ejpam-4396	108	78	for	for	ADP
ejpam-4396	108	79	all	all	DET
ejpam-4396	108	80	x	x	SYM
ejpam-4396	108	81	∈	∈	PROPN
ejpam-4396	108	82	g	g	NOUN
ejpam-4396	108	83	,	,	PUNCT
ejpam-4396	108	84	we	we	PRON
ejpam-4396	108	85	have	have	VERB
ejpam-4396	108	86	e	e	NOUN
ejpam-4396	108	87	∗	∗	NOUN
ejpam-4396	108	88	x	x	X
ejpam-4396	109	1	=	=	PUNCT
ejpam-4396	109	2	x	x	SYM
ejpam-4396	109	3	=	=	PUNCT
ejpam-4396	109	4	x	x	X
ejpam-4396	109	5	∗	∗	X
ejpam-4396	109	6	e	e	NOUN
ejpam-4396	109	7	;	;	PUNCT
ejpam-4396	109	8	and	and	CCONJ
ejpam-4396	109	9	,	,	PUNCT
ejpam-4396	109	10	(	(	PUNCT
ejpam-4396	109	11	g3	g3	NOUN
ejpam-4396	109	12	)	)	PUNCT
ejpam-4396	109	13	for	for	ADP
ejpam-4396	109	14	each	each	PRON
ejpam-4396	109	15	a	a	DET
ejpam-4396	109	16	∈	∈	PROPN
ejpam-4396	109	17	g	g	NOUN
ejpam-4396	109	18	,	,	PUNCT
ejpam-4396	109	19	there	there	PRON
ejpam-4396	109	20	exists	exist	VERB
ejpam-4396	109	21	an	an	DET
ejpam-4396	109	22	element	element	NOUN
ejpam-4396	109	23	b	b	NOUN
ejpam-4396	109	24	in	in	ADP
ejpam-4396	109	25	g	g	PROPN
ejpam-4396	109	26	such	such	DET
ejpam-4396	109	27	that	that	SCONJ
ejpam-4396	109	28	a	a	DET
ejpam-4396	109	29	∗	∗	NOUN
ejpam-4396	109	30	b	b	NOUN
ejpam-4396	109	31	=	=	SYM
ejpam-4396	109	32	e	e	NOUN
ejpam-4396	109	33	=	=	SYM
ejpam-4396	109	34	b	b	PROPN
ejpam-4396	109	35	∗	∗	X
ejpam-4396	109	36	a.	a.	NOUN
ejpam-4396	109	37	corollary	corollary	PROPN
ejpam-4396	109	38	1	1	PROPN
ejpam-4396	109	39	.	.	PUNCT
ejpam-4396	110	1	let	let	VERB
ejpam-4396	110	2	g	g	NOUN
ejpam-4396	110	3	be	be	AUX
ejpam-4396	110	4	an	an	DET
ejpam-4396	110	5	abelian	abelian	ADJ
ejpam-4396	110	6	g	g	NOUN
ejpam-4396	110	7	-	-	PUNCT
ejpam-4396	110	8	group	group	NOUN
ejpam-4396	110	9	.	.	PUNCT
ejpam-4396	111	1	if	if	SCONJ
ejpam-4396	111	2	h	h	NOUN
ejpam-4396	111	3	=	=	NOUN
ejpam-4396	111	4	∅	∅	NOUN
ejpam-4396	111	5	,	,	PUNCT
ejpam-4396	111	6	then	then	ADV
ejpam-4396	111	7	h	h	PROPN
ejpam-4396	111	8	is	be	AUX
ejpam-4396	111	9	a	a	DET
ejpam-4396	111	10	group	group	NOUN
ejpam-4396	111	11	.	.	PUNCT
ejpam-4396	112	1	j.	j.	PROPN
ejpam-4396	112	2	caraquil	caraquil	PROPN
ejpam-4396	112	3	,	,	PUNCT
ejpam-4396	112	4	m.	m.	PROPN
ejpam-4396	112	5	baldado	baldado	PROPN
ejpam-4396	112	6	jr	jr	PROPN
ejpam-4396	112	7	.	.	PROPN
ejpam-4396	112	8	/	/	SYM
ejpam-4396	112	9	eur	eur	PROPN
ejpam-4396	112	10	.	.	PUNCT
ejpam-4396	113	1	j.	j.	PROPN
ejpam-4396	113	2	pure	pure	PROPN
ejpam-4396	113	3	appl	appl	PROPN
ejpam-4396	113	4	.	.	PROPN
ejpam-4396	113	5	math	math	PROPN
ejpam-4396	113	6	,	,	PUNCT
ejpam-4396	113	7	15	15	NUM
ejpam-4396	113	8	(	(	PUNCT
ejpam-4396	113	9	3	3	NUM
ejpam-4396	113	10	)	)	PUNCT
ejpam-4396	113	11	(	(	PUNCT
ejpam-4396	113	12	2022	2022	NUM
ejpam-4396	113	13	)	)	PUNCT
ejpam-4396	113	14	,	,	PUNCT
ejpam-4396	113	15	887	887	NUM
ejpam-4396	113	16	-	-	SYM
ejpam-4396	113	17	896	896	NUM
ejpam-4396	113	18	891	891	NUM
ejpam-4396	113	19	proof	proof	NOUN
ejpam-4396	113	20	.	.	PUNCT
ejpam-4396	114	1	(	(	PUNCT
ejpam-4396	114	2	g1	g1	PROPN
ejpam-4396	114	3	)	)	PUNCT
ejpam-4396	114	4	is	be	AUX
ejpam-4396	114	5	satisfied	satisfied	ADJ
ejpam-4396	114	6	by	by	ADP
ejpam-4396	114	7	the	the	DET
ejpam-4396	114	8	fact	fact	NOUN
ejpam-4396	114	9	that	that	SCONJ
ejpam-4396	114	10	h	h	NOUN
ejpam-4396	114	11	is	be	AUX
ejpam-4396	114	12	a	a	DET
ejpam-4396	114	13	subset	subset	NOUN
ejpam-4396	114	14	of	of	ADP
ejpam-4396	114	15	g.	g.	PROPN
ejpam-4396	114	16	(	(	PUNCT
ejpam-4396	114	17	g2	g2	PROPN
ejpam-4396	114	18	)	)	PUNCT
ejpam-4396	114	19	follows	follow	VERB
ejpam-4396	114	20	from	from	ADP
ejpam-4396	114	21	lemma	lemma	PROPN
ejpam-4396	114	22	1	1	NUM
ejpam-4396	114	23	,	,	PUNCT
ejpam-4396	114	24	and	and	CCONJ
ejpam-4396	114	25	(	(	PUNCT
ejpam-4396	114	26	g3	g3	NOUN
ejpam-4396	114	27	)	)	PUNCT
ejpam-4396	114	28	follows	follow	VERB
ejpam-4396	114	29	from	from	ADP
ejpam-4396	114	30	remark	remark	NOUN
ejpam-4396	114	31	1	1	NUM
ejpam-4396	114	32	and	and	CCONJ
ejpam-4396	114	33	remark	remark	NOUN
ejpam-4396	114	34	2	2	NUM
ejpam-4396	114	35	.	.	PUNCT
ejpam-4396	115	1	in	in	ADP
ejpam-4396	115	2	the	the	DET
ejpam-4396	115	3	same	same	ADJ
ejpam-4396	115	4	sense	sense	NOUN
ejpam-4396	115	5	as	as	ADP
ejpam-4396	115	6	in	in	ADP
ejpam-4396	115	7	the	the	DET
ejpam-4396	115	8	proof	proof	NOUN
ejpam-4396	115	9	of	of	ADP
ejpam-4396	115	10	theorem	theorem	NOUN
ejpam-4396	115	11	1	1	NUM
ejpam-4396	115	12	the	the	DET
ejpam-4396	115	13	operation	operation	NOUN
ejpam-4396	115	14	can	can	AUX
ejpam-4396	115	15	be	be	AUX
ejpam-4396	115	16	shown	show	VERB
ejpam-4396	115	17	to	to	PART
ejpam-4396	115	18	be	be	AUX
ejpam-4396	115	19	a	a	DET
ejpam-4396	115	20	binary	binary	ADJ
ejpam-4396	115	21	operation	operation	NOUN
ejpam-4396	115	22	in	in	ADP
ejpam-4396	115	23	h.	h.	PROPN
ejpam-4396	115	24	the	the	DET
ejpam-4396	115	25	next	next	ADJ
ejpam-4396	115	26	statement	statement	NOUN
ejpam-4396	115	27	,	,	PUNCT
ejpam-4396	115	28	lemma	lemma	PROPN
ejpam-4396	115	29	2	2	NUM
ejpam-4396	115	30	,	,	PUNCT
ejpam-4396	115	31	says	say	VERB
ejpam-4396	115	32	that	that	SCONJ
ejpam-4396	115	33	g\h	g\h	PROPN
ejpam-4396	115	34	is	be	AUX
ejpam-4396	115	35	closed	close	VERB
ejpam-4396	115	36	with	with	ADP
ejpam-4396	115	37	respect	respect	NOUN
ejpam-4396	115	38	to	to	ADP
ejpam-4396	115	39	the	the	DET
ejpam-4396	115	40	binary	binary	ADJ
ejpam-4396	115	41	operation	operation	NOUN
ejpam-4396	115	42	in	in	ADP
ejpam-4396	115	43	g.	g.	PROPN
ejpam-4396	115	44	lemma	lemma	PROPN
ejpam-4396	116	1	2	2	X
ejpam-4396	116	2	.	.	PUNCT
ejpam-4396	116	3	let	let	VERB
ejpam-4396	116	4	g	g	NOUN
ejpam-4396	116	5	be	be	AUX
ejpam-4396	116	6	an	an	DET
ejpam-4396	116	7	abelian	abelian	ADJ
ejpam-4396	116	8	g	g	NOUN
ejpam-4396	116	9	-	-	PUNCT
ejpam-4396	116	10	group	group	NOUN
ejpam-4396	116	11	.	.	PUNCT
ejpam-4396	117	1	if	if	SCONJ
ejpam-4396	117	2	a	a	DET
ejpam-4396	117	3	∈	∈	NOUN
ejpam-4396	117	4	g\h	g\h	PROPN
ejpam-4396	117	5	,	,	PUNCT
ejpam-4396	117	6	then	then	ADV
ejpam-4396	117	7	ab	ab	PROPN
ejpam-4396	117	8	∈	∈	PROPN
ejpam-4396	117	9	g\h	g\h	PROPN
ejpam-4396	117	10	for	for	ADP
ejpam-4396	117	11	all	all	DET
ejpam-4396	117	12	b	b	PROPN
ejpam-4396	117	13	∈	∈	PROPN
ejpam-4396	117	14	g.	g.	NOUN
ejpam-4396	117	15	proof	proof	NOUN
ejpam-4396	117	16	.	.	PUNCT
ejpam-4396	118	1	let	let	VERB
ejpam-4396	118	2	a	a	DET
ejpam-4396	118	3	∈	∈	NOUN
ejpam-4396	118	4	g\h	g\h	PROPN
ejpam-4396	118	5	and	and	CCONJ
ejpam-4396	118	6	ab	ab	PROPN
ejpam-4396	118	7	∈	∈	PROPN
ejpam-4396	118	8	h.	h.	NOUN
ejpam-4396	119	1	if	if	SCONJ
ejpam-4396	119	2	a	a	DET
ejpam-4396	119	3	∈	∈	NOUN
ejpam-4396	119	4	g\h	g\h	PROPN
ejpam-4396	119	5	,	,	PUNCT
ejpam-4396	119	6	then	then	ADV
ejpam-4396	119	7	there	there	PRON
ejpam-4396	119	8	exist	exist	VERB
ejpam-4396	119	9	e	e	NOUN
ejpam-4396	119	10	and	and	CCONJ
ejpam-4396	119	11	e′	e′	NOUN
ejpam-4396	119	12	such	such	ADJ
ejpam-4396	119	13	that	that	PRON
ejpam-4396	119	14	ea	ea	NOUN
ejpam-4396	119	15	=	=	PUNCT
ejpam-4396	119	16	a	a	PROPN
ejpam-4396	119	17	=	=	X
ejpam-4396	119	18	e′a	e′a	PROPN
ejpam-4396	119	19	where	where	SCONJ
ejpam-4396	119	20	e	e	X
ejpam-4396	119	21	̸=	̸=	PROPN
ejpam-4396	119	22	e′.	e′.	NOUN
ejpam-4396	119	23	hence	hence	ADV
ejpam-4396	119	24	,	,	PUNCT
ejpam-4396	119	25	eab	eab	NOUN
ejpam-4396	119	26	=	=	SYM
ejpam-4396	119	27	ab	ab	PROPN
ejpam-4396	119	28	=	=	PUNCT
ejpam-4396	119	29	e′ab	e′ab	NOUN
ejpam-4396	119	30	.	.	PUNCT
ejpam-4396	120	1	since	since	SCONJ
ejpam-4396	120	2	ab	ab	PROPN
ejpam-4396	120	3	is	be	AUX
ejpam-4396	120	4	a	a	DET
ejpam-4396	120	5	unit	unit	NOUN
ejpam-4396	120	6	,	,	PUNCT
ejpam-4396	120	7	we	we	PRON
ejpam-4396	120	8	must	must	AUX
ejpam-4396	120	9	have	have	VERB
ejpam-4396	120	10	e	e	NOUN
ejpam-4396	120	11	=	=	NOUN
ejpam-4396	120	12	e′.	e′.	NUM
ejpam-4396	120	13	this	this	PRON
ejpam-4396	120	14	is	be	AUX
ejpam-4396	120	15	a	a	DET
ejpam-4396	120	16	contradiction	contradiction	NOUN
ejpam-4396	120	17	.	.	PUNCT
ejpam-4396	121	1	the	the	DET
ejpam-4396	121	2	next	next	ADJ
ejpam-4396	121	3	statement	statement	NOUN
ejpam-4396	121	4	,	,	PUNCT
ejpam-4396	121	5	corollary	corollary	ADJ
ejpam-4396	121	6	2	2	NUM
ejpam-4396	121	7	,	,	PUNCT
ejpam-4396	121	8	follows	follow	VERB
ejpam-4396	121	9	directly	directly	ADV
ejpam-4396	121	10	from	from	ADP
ejpam-4396	121	11	lemma	lemma	PROPN
ejpam-4396	121	12	2	2	NUM
ejpam-4396	121	13	.	.	PUNCT
ejpam-4396	121	14	corollary	corollary	ADJ
ejpam-4396	121	15	2	2	NUM
ejpam-4396	121	16	.	.	PUNCT
ejpam-4396	122	1	let	let	VERB
ejpam-4396	122	2	g	g	NOUN
ejpam-4396	122	3	be	be	AUX
ejpam-4396	122	4	an	an	DET
ejpam-4396	122	5	abelian	abelian	ADJ
ejpam-4396	122	6	g	g	NOUN
ejpam-4396	122	7	-	-	PUNCT
ejpam-4396	122	8	group	group	NOUN
ejpam-4396	122	9	.	.	PUNCT
ejpam-4396	123	1	if	if	SCONJ
ejpam-4396	123	2	a	a	DET
ejpam-4396	123	3	,	,	PUNCT
ejpam-4396	123	4	b	b	NOUN
ejpam-4396	123	5	∈	∈	PROPN
ejpam-4396	123	6	g\h	g\h	PROPN
ejpam-4396	123	7	,	,	PUNCT
ejpam-4396	123	8	then	then	ADV
ejpam-4396	123	9	ab	ab	PROPN
ejpam-4396	123	10	∈	∈	PROPN
ejpam-4396	123	11	g\h	g\h	PROPN
ejpam-4396	123	12	.	.	PUNCT
ejpam-4396	124	1	the	the	DET
ejpam-4396	124	2	next	next	ADJ
ejpam-4396	124	3	statement	statement	NOUN
ejpam-4396	124	4	,	,	PUNCT
ejpam-4396	124	5	lemma	lemma	PROPN
ejpam-4396	124	6	3	3	NUM
ejpam-4396	124	7	,	,	PUNCT
ejpam-4396	124	8	says	say	VERB
ejpam-4396	124	9	that	that	SCONJ
ejpam-4396	124	10	every	every	DET
ejpam-4396	124	11	element	element	NOUN
ejpam-4396	124	12	in	in	ADP
ejpam-4396	124	13	g\h	g\h	PROPN
ejpam-4396	124	14	has	have	VERB
ejpam-4396	124	15	an	an	DET
ejpam-4396	124	16	identity	identity	NOUN
ejpam-4396	124	17	element	element	NOUN
ejpam-4396	124	18	in	in	ADP
ejpam-4396	124	19	g\h	g\h	PROPN
ejpam-4396	124	20	,	,	PUNCT
ejpam-4396	124	21	that	that	ADV
ejpam-4396	124	22	is	is	ADV
ejpam-4396	124	23	,	,	PUNCT
ejpam-4396	124	24	g\h	g\h	PROPN
ejpam-4396	124	25	satisfies	satisfy	VERB
ejpam-4396	124	26	(	(	PUNCT
ejpam-4396	124	27	g2	g2	PROPN
ejpam-4396	124	28	)	)	PUNCT
ejpam-4396	124	29	.	.	PUNCT
ejpam-4396	125	1	lemma	lemma	PROPN
ejpam-4396	125	2	3	3	X
ejpam-4396	125	3	.	.	PUNCT
ejpam-4396	126	1	let	let	VERB
ejpam-4396	126	2	g	g	NOUN
ejpam-4396	126	3	be	be	AUX
ejpam-4396	126	4	an	an	DET
ejpam-4396	126	5	abelian	abelian	ADJ
ejpam-4396	126	6	g	g	NOUN
ejpam-4396	126	7	-	-	PUNCT
ejpam-4396	126	8	group	group	NOUN
ejpam-4396	126	9	.	.	PUNCT
ejpam-4396	127	1	if	if	SCONJ
ejpam-4396	127	2	a	a	DET
ejpam-4396	127	3	∈	∈	NOUN
ejpam-4396	127	4	g\h	g\h	PROPN
ejpam-4396	127	5	,	,	PUNCT
ejpam-4396	127	6	then	then	ADV
ejpam-4396	127	7	a	a	PRON
ejpam-4396	127	8	has	have	VERB
ejpam-4396	127	9	an	an	DET
ejpam-4396	127	10	identity	identity	NOUN
ejpam-4396	127	11	in	in	ADP
ejpam-4396	127	12	g\h	g\h	PROPN
ejpam-4396	127	13	.	.	PUNCT
ejpam-4396	128	1	proof	proof	NOUN
ejpam-4396	128	2	.	.	PUNCT
ejpam-4396	129	1	if	if	SCONJ
ejpam-4396	129	2	a	a	DET
ejpam-4396	129	3	∈	∈	NOUN
ejpam-4396	129	4	g\h	g\h	PROPN
ejpam-4396	129	5	,	,	PUNCT
ejpam-4396	129	6	then	then	ADV
ejpam-4396	129	7	a	a	PRON
ejpam-4396	129	8	has	have	VERB
ejpam-4396	129	9	two	two	NUM
ejpam-4396	129	10	or	or	CCONJ
ejpam-4396	129	11	more	more	ADJ
ejpam-4396	129	12	identity	identity	NOUN
ejpam-4396	129	13	elements	element	NOUN
ejpam-4396	129	14	,	,	PUNCT
ejpam-4396	129	15	say	say	VERB
ejpam-4396	129	16	e	e	NOUN
ejpam-4396	129	17	and	and	CCONJ
ejpam-4396	129	18	e′	e′	NOUN
ejpam-4396	129	19	are	be	AUX
ejpam-4396	129	20	two	two	NUM
ejpam-4396	129	21	of	of	ADP
ejpam-4396	129	22	its	its	PRON
ejpam-4396	129	23	distinct	distinct	ADJ
ejpam-4396	129	24	identities	identity	NOUN
ejpam-4396	129	25	.	.	PUNCT
ejpam-4396	130	1	by	by	ADP
ejpam-4396	130	2	remark	remark	NOUN
ejpam-4396	130	3	5	5	NUM
ejpam-4396	130	4	a	a	PRON
ejpam-4396	130	5	has	have	VERB
ejpam-4396	130	6	a	a	DET
ejpam-4396	130	7	unique	unique	ADJ
ejpam-4396	130	8	identity	identity	NOUN
ejpam-4396	130	9	,	,	PUNCT
ejpam-4396	130	10	say	say	VERB
ejpam-4396	130	11	ea	ea	INTJ
ejpam-4396	130	12	,	,	PUNCT
ejpam-4396	130	13	such	such	ADJ
ejpam-4396	130	14	that	that	DET
ejpam-4396	130	15	aa−1	aa−1	PROPN
ejpam-4396	130	16	=	=	SYM
ejpam-4396	130	17	ea	ea	PROPN
ejpam-4396	130	18	.	.	PUNCT
ejpam-4396	130	19	note	note	VERB
ejpam-4396	130	20	that	that	SCONJ
ejpam-4396	130	21	eae	eae	PROPN
ejpam-4396	130	22	=	=	PUNCT
ejpam-4396	130	23	a−1ae	a−1ae	PROPN
ejpam-4396	130	24	=	=	SYM
ejpam-4396	130	25	a−1a	a−1a	PROPN
ejpam-4396	130	26	=	=	SYM
ejpam-4396	130	27	ea	ea	PROPN
ejpam-4396	130	28	,	,	PUNCT
ejpam-4396	130	29	that	that	PRON
ejpam-4396	130	30	is	is	ADV
ejpam-4396	130	31	e	e	NOUN
ejpam-4396	130	32	is	be	AUX
ejpam-4396	130	33	an	an	DET
ejpam-4396	130	34	identity	identity	NOUN
ejpam-4396	130	35	of	of	ADP
ejpam-4396	130	36	ea	ea	PROPN
ejpam-4396	130	37	.	.	PUNCT
ejpam-4396	131	1	similarly	similarly	ADV
ejpam-4396	131	2	,	,	PUNCT
ejpam-4396	131	3	note	note	VERB
ejpam-4396	131	4	that	that	SCONJ
ejpam-4396	131	5	eae	eae	PROPN
ejpam-4396	131	6	′	′	NOUN
ejpam-4396	132	1	=	=	PUNCT
ejpam-4396	133	1	a−1ae′	a−1ae′	NUM
ejpam-4396	133	2	=	=	PUNCT
ejpam-4396	133	3	a−1a	a−1a	NOUN
ejpam-4396	133	4	=	=	SYM
ejpam-4396	133	5	ea	ea	PROPN
ejpam-4396	133	6	,	,	PUNCT
ejpam-4396	133	7	that	that	PRON
ejpam-4396	133	8	is	be	AUX
ejpam-4396	133	9	e′	e′	NOUN
ejpam-4396	133	10	is	be	AUX
ejpam-4396	133	11	an	an	DET
ejpam-4396	133	12	identity	identity	NOUN
ejpam-4396	133	13	of	of	ADP
ejpam-4396	133	14	ea	ea	NOUN
ejpam-4396	133	15	.	.	PUNCT
ejpam-4396	134	1	since	since	SCONJ
ejpam-4396	134	2	e	e	PROPN
ejpam-4396	134	3	̸=	̸=	PROPN
ejpam-4396	134	4	e′	e′	NUM
ejpam-4396	134	5	,	,	PUNCT
ejpam-4396	134	6	ea	ea	PROPN
ejpam-4396	134	7	is	be	AUX
ejpam-4396	134	8	not	not	PART
ejpam-4396	134	9	a	a	DET
ejpam-4396	134	10	unit	unit	NOUN
ejpam-4396	134	11	,	,	PUNCT
ejpam-4396	134	12	that	that	PRON
ejpam-4396	134	13	is	be	AUX
ejpam-4396	134	14	ea	ea	NOUN
ejpam-4396	134	15	∈	∈	PROPN
ejpam-4396	134	16	g\h	g\h	NOUN
ejpam-4396	134	17	.	.	PUNCT
ejpam-4396	135	1	the	the	DET
ejpam-4396	135	2	next	next	ADJ
ejpam-4396	135	3	statement	statement	NOUN
ejpam-4396	135	4	,	,	PUNCT
ejpam-4396	135	5	lemma	lemma	PROPN
ejpam-4396	135	6	4	4	NUM
ejpam-4396	135	7	,	,	PUNCT
ejpam-4396	135	8	says	say	VERB
ejpam-4396	135	9	that	that	SCONJ
ejpam-4396	135	10	inverse	inverse	NOUN
ejpam-4396	135	11	of	of	ADP
ejpam-4396	135	12	a	a	DET
ejpam-4396	135	13	non	non	ADJ
ejpam-4396	135	14	-	-	NOUN
ejpam-4396	135	15	unit	unit	NOUN
ejpam-4396	135	16	is	be	AUX
ejpam-4396	135	17	a	a	DET
ejpam-4396	135	18	non	non	ADJ
ejpam-4396	135	19	-	-	NOUN
ejpam-4396	135	20	unit	unit	NOUN
ejpam-4396	135	21	,	,	PUNCT
ejpam-4396	135	22	that	that	ADV
ejpam-4396	135	23	is	is	ADV
ejpam-4396	135	24	,	,	PUNCT
ejpam-4396	135	25	g\h	g\h	PROPN
ejpam-4396	135	26	satisfies	satisfie	NOUN
ejpam-4396	135	27	(	(	PUNCT
ejpam-4396	135	28	g3	g3	PROPN
ejpam-4396	135	29	)	)	PUNCT
ejpam-4396	135	30	.	.	PUNCT
ejpam-4396	136	1	lemma	lemma	PROPN
ejpam-4396	136	2	4	4	X
ejpam-4396	136	3	.	.	PUNCT
ejpam-4396	137	1	let	let	VERB
ejpam-4396	137	2	g	g	NOUN
ejpam-4396	137	3	be	be	AUX
ejpam-4396	137	4	an	an	DET
ejpam-4396	137	5	abelian	abelian	ADJ
ejpam-4396	137	6	g	g	NOUN
ejpam-4396	137	7	-	-	PUNCT
ejpam-4396	137	8	group	group	NOUN
ejpam-4396	137	9	.	.	PUNCT
ejpam-4396	138	1	if	if	SCONJ
ejpam-4396	138	2	a	a	DET
ejpam-4396	138	3	∈	∈	NOUN
ejpam-4396	138	4	g\h	g\h	PROPN
ejpam-4396	138	5	,	,	PUNCT
ejpam-4396	138	6	then	then	ADV
ejpam-4396	138	7	a	a	PRON
ejpam-4396	138	8	has	have	VERB
ejpam-4396	138	9	an	an	DET
ejpam-4396	138	10	inverse	inverse	NOUN
ejpam-4396	138	11	g\h	g\h	NOUN
ejpam-4396	138	12	.	.	PUNCT
ejpam-4396	139	1	proof	proof	NOUN
ejpam-4396	139	2	.	.	PUNCT
ejpam-4396	140	1	if	if	SCONJ
ejpam-4396	140	2	a	a	DET
ejpam-4396	140	3	∈	∈	NOUN
ejpam-4396	140	4	g\h	g\h	NOUN
ejpam-4396	140	5	,	,	PUNCT
ejpam-4396	140	6	then	then	ADV
ejpam-4396	140	7	by	by	ADP
ejpam-4396	140	8	remark	remark	NOUN
ejpam-4396	140	9	5	5	NUM
ejpam-4396	140	10	,	,	PUNCT
ejpam-4396	140	11	a	a	PRON
ejpam-4396	140	12	has	have	VERB
ejpam-4396	140	13	a	a	DET
ejpam-4396	140	14	unique	unique	ADJ
ejpam-4396	140	15	identity	identity	NOUN
ejpam-4396	140	16	such	such	ADJ
ejpam-4396	140	17	that	that	SCONJ
ejpam-4396	140	18	a	a	PRON
ejpam-4396	140	19	has	have	VERB
ejpam-4396	140	20	an	an	DET
ejpam-4396	140	21	inverse	inverse	NOUN
ejpam-4396	140	22	,	,	PUNCT
ejpam-4396	140	23	say	say	VERB
ejpam-4396	140	24	the	the	DET
ejpam-4396	140	25	inverse	inverse	NOUN
ejpam-4396	140	26	is	be	AUX
ejpam-4396	140	27	b.	b.	NOUN
ejpam-4396	140	28	suppose	suppose	VERB
ejpam-4396	140	29	that	that	SCONJ
ejpam-4396	140	30	b	b	PROPN
ejpam-4396	140	31	∈	∈	PROPN
ejpam-4396	140	32	h.	h.	NOUN
ejpam-4396	140	33	then	then	ADV
ejpam-4396	140	34	by	by	ADP
ejpam-4396	140	35	remark	remark	NOUN
ejpam-4396	140	36	2	2	NUM
ejpam-4396	140	37	,	,	PUNCT
ejpam-4396	140	38	b	b	NOUN
ejpam-4396	140	39	has	have	VERB
ejpam-4396	140	40	a	a	DET
ejpam-4396	140	41	unique	unique	ADJ
ejpam-4396	140	42	inverse	inverse	NOUN
ejpam-4396	140	43	,	,	PUNCT
ejpam-4396	140	44	which	which	PRON
ejpam-4396	140	45	by	by	ADP
ejpam-4396	140	46	remark	remark	NOUN
ejpam-4396	140	47	1	1	NUM
ejpam-4396	140	48	must	must	AUX
ejpam-4396	140	49	be	be	AUX
ejpam-4396	140	50	in	in	ADP
ejpam-4396	140	51	h.	h.	PROPN
ejpam-4396	140	52	this	this	PRON
ejpam-4396	140	53	is	be	AUX
ejpam-4396	140	54	a	a	DET
ejpam-4396	140	55	contradiction	contradiction	NOUN
ejpam-4396	140	56	since	since	SCONJ
ejpam-4396	140	57	a	a	PRON
ejpam-4396	140	58	(	(	PUNCT
ejpam-4396	140	59	which	which	PRON
ejpam-4396	140	60	is	be	AUX
ejpam-4396	140	61	in	in	ADP
ejpam-4396	140	62	g\h	g\h	NOUN
ejpam-4396	140	63	)	)	PUNCT
ejpam-4396	140	64	is	be	AUX
ejpam-4396	140	65	also	also	ADV
ejpam-4396	140	66	an	an	DET
ejpam-4396	140	67	inverse	inverse	NOUN
ejpam-4396	140	68	of	of	ADP
ejpam-4396	140	69	b.	b.	PROPN
ejpam-4396	140	70	finally	finally	ADV
ejpam-4396	140	71	,	,	PUNCT
ejpam-4396	140	72	the	the	DET
ejpam-4396	140	73	next	next	ADJ
ejpam-4396	140	74	statement	statement	NOUN
ejpam-4396	140	75	,	,	PUNCT
ejpam-4396	140	76	theorem	theorem	VERB
ejpam-4396	140	77	2	2	NUM
ejpam-4396	140	78	,	,	PUNCT
ejpam-4396	140	79	provides	provide	VERB
ejpam-4396	140	80	a	a	DET
ejpam-4396	140	81	way	way	NOUN
ejpam-4396	140	82	of	of	ADP
ejpam-4396	140	83	constructing	construct	VERB
ejpam-4396	140	84	a	a	DET
ejpam-4396	140	85	g	g	NOUN
ejpam-4396	140	86	-	-	PUNCT
ejpam-4396	140	87	subgroup	subgroup	NOUN
ejpam-4396	140	88	.	.	PUNCT
ejpam-4396	141	1	it	it	PRON
ejpam-4396	141	2	says	say	VERB
ejpam-4396	141	3	that	that	SCONJ
ejpam-4396	141	4	if	if	SCONJ
ejpam-4396	141	5	h	h	NOUN
ejpam-4396	141	6	is	be	AUX
ejpam-4396	141	7	the	the	DET
ejpam-4396	141	8	set	set	NOUN
ejpam-4396	141	9	of	of	ADP
ejpam-4396	141	10	all	all	DET
ejpam-4396	141	11	units	unit	NOUN
ejpam-4396	141	12	,	,	PUNCT
ejpam-4396	141	13	then	then	ADV
ejpam-4396	141	14	its	its	PRON
ejpam-4396	141	15	complement	complement	NOUN
ejpam-4396	141	16	is	be	AUX
ejpam-4396	141	17	a	a	DET
ejpam-4396	141	18	g	g	NOUN
ejpam-4396	141	19	-	-	PUNCT
ejpam-4396	141	20	subgroup	subgroup	NOUN
ejpam-4396	141	21	also	also	ADV
ejpam-4396	141	22	.	.	PUNCT
ejpam-4396	142	1	theorem	theorem	ADJ
ejpam-4396	142	2	2	2	X
ejpam-4396	142	3	.	.	PUNCT
ejpam-4396	143	1	let	let	VERB
ejpam-4396	143	2	g	g	NOUN
ejpam-4396	143	3	be	be	AUX
ejpam-4396	143	4	an	an	DET
ejpam-4396	143	5	abelian	abelian	ADJ
ejpam-4396	143	6	g	g	NOUN
ejpam-4396	143	7	-	-	PUNCT
ejpam-4396	143	8	group	group	NOUN
ejpam-4396	143	9	.	.	PUNCT
ejpam-4396	144	1	if	if	SCONJ
ejpam-4396	144	2	h	h	NOUN
ejpam-4396	144	3	̸=	̸=	PROPN
ejpam-4396	144	4	g	g	NOUN
ejpam-4396	144	5	,	,	PUNCT
ejpam-4396	144	6	then	then	ADV
ejpam-4396	144	7	g\h	g\h	PROPN
ejpam-4396	144	8	is	be	AUX
ejpam-4396	144	9	a	a	DET
ejpam-4396	144	10	g	g	NOUN
ejpam-4396	144	11	-	-	PUNCT
ejpam-4396	144	12	subgroup	subgroup	NOUN
ejpam-4396	144	13	of	of	ADP
ejpam-4396	144	14	g.	g.	PROPN
ejpam-4396	144	15	proof	proof	NOUN
ejpam-4396	144	16	.	.	PUNCT
ejpam-4396	145	1	by	by	ADP
ejpam-4396	145	2	corollary	corollary	ADJ
ejpam-4396	145	3	2	2	NUM
ejpam-4396	145	4	,	,	PUNCT
ejpam-4396	145	5	the	the	DET
ejpam-4396	145	6	operation	operation	NOUN
ejpam-4396	145	7	in	in	ADP
ejpam-4396	145	8	g	g	PROPN
ejpam-4396	145	9	is	be	AUX
ejpam-4396	145	10	a	a	DET
ejpam-4396	145	11	binary	binary	ADJ
ejpam-4396	145	12	operation	operation	NOUN
ejpam-4396	145	13	in	in	ADP
ejpam-4396	145	14	g\h	g\h	PROPN
ejpam-4396	145	15	.	.	PUNCT
ejpam-4396	146	1	(	(	PUNCT
ejpam-4396	146	2	g1	g1	PROPN
ejpam-4396	146	3	)	)	PUNCT
ejpam-4396	146	4	follows	follow	VERB
ejpam-4396	146	5	from	from	ADP
ejpam-4396	146	6	the	the	DET
ejpam-4396	146	7	fact	fact	NOUN
ejpam-4396	146	8	that	that	SCONJ
ejpam-4396	146	9	g\h	g\h	PROPN
ejpam-4396	146	10	is	be	AUX
ejpam-4396	146	11	a	a	DET
ejpam-4396	146	12	subset	subset	NOUN
ejpam-4396	146	13	of	of	ADP
ejpam-4396	146	14	g	g	PROPN
ejpam-4396	146	15	the	the	DET
ejpam-4396	146	16	operation	operation	NOUN
ejpam-4396	146	17	is	be	AUX
ejpam-4396	146	18	associative	associative	ADJ
ejpam-4396	146	19	in	in	ADP
ejpam-4396	146	20	g.	g.	PROPN
ejpam-4396	146	21	(	(	PUNCT
ejpam-4396	146	22	g2	g2	PROPN
ejpam-4396	146	23	)	)	PUNCT
ejpam-4396	146	24	follows	follow	VERB
ejpam-4396	146	25	from	from	ADP
ejpam-4396	146	26	lemma	lemma	PROPN
ejpam-4396	146	27	3	3	NUM
ejpam-4396	146	28	,	,	PUNCT
ejpam-4396	146	29	while	while	SCONJ
ejpam-4396	146	30	(	(	PUNCT
ejpam-4396	146	31	g3	g3	NOUN
ejpam-4396	146	32	)	)	PUNCT
ejpam-4396	146	33	follows	follow	VERB
ejpam-4396	146	34	from	from	ADP
ejpam-4396	146	35	lemma	lemma	PROPN
ejpam-4396	146	36	4	4	NUM
ejpam-4396	146	37	.	.	PUNCT
ejpam-4396	147	1	theorem	theorem	VERB
ejpam-4396	147	2	1	1	NUM
ejpam-4396	147	3	and	and	CCONJ
ejpam-4396	147	4	theorem	theorem	VERB
ejpam-4396	147	5	2	2	NUM
ejpam-4396	147	6	implies	imply	VERB
ejpam-4396	147	7	that	that	SCONJ
ejpam-4396	147	8	g	g	NOUN
ejpam-4396	147	9	-	-	PUNCT
ejpam-4396	147	10	groups	group	NOUN
ejpam-4396	147	11	may	may	AUX
ejpam-4396	147	12	be	be	AUX
ejpam-4396	147	13	partitioned	partition	VERB
ejpam-4396	147	14	into	into	ADP
ejpam-4396	147	15	g	g	NOUN
ejpam-4396	147	16	-	-	PUNCT
ejpam-4396	147	17	subgroups	subgroup	NOUN
ejpam-4396	147	18	.	.	PUNCT
ejpam-4396	148	1	in	in	ADP
ejpam-4396	148	2	particular	particular	ADJ
ejpam-4396	148	3	,	,	PUNCT
ejpam-4396	148	4	the	the	DET
ejpam-4396	148	5	set	set	NOUN
ejpam-4396	148	6	of	of	ADP
ejpam-4396	148	7	all	all	DET
ejpam-4396	148	8	units	unit	NOUN
ejpam-4396	148	9	of	of	ADP
ejpam-4396	148	10	a	a	DET
ejpam-4396	148	11	g	g	NOUN
ejpam-4396	148	12	-	-	PUNCT
ejpam-4396	148	13	group	group	NOUN
ejpam-4396	148	14	and	and	CCONJ
ejpam-4396	148	15	its	its	PRON
ejpam-4396	148	16	complement	complement	NOUN
ejpam-4396	148	17	are	be	AUX
ejpam-4396	148	18	both	both	PRON
ejpam-4396	148	19	g	g	NOUN
ejpam-4396	148	20	-	-	PUNCT
ejpam-4396	148	21	subgroups	subgroup	NOUN
ejpam-4396	148	22	.	.	PUNCT
ejpam-4396	149	1	this	this	PRON
ejpam-4396	149	2	is	be	AUX
ejpam-4396	149	3	not	not	PART
ejpam-4396	149	4	the	the	DET
ejpam-4396	149	5	case	case	NOUN
ejpam-4396	149	6	for	for	ADP
ejpam-4396	149	7	groups	group	NOUN
ejpam-4396	149	8	.	.	PUNCT
ejpam-4396	150	1	j.	j.	PROPN
ejpam-4396	150	2	caraquil	caraquil	PROPN
ejpam-4396	150	3	,	,	PUNCT
ejpam-4396	150	4	m.	m.	PROPN
ejpam-4396	150	5	baldado	baldado	PROPN
ejpam-4396	150	6	jr	jr	PROPN
ejpam-4396	150	7	.	.	PROPN
ejpam-4396	150	8	/	/	SYM
ejpam-4396	150	9	eur	eur	PROPN
ejpam-4396	150	10	.	.	PUNCT
ejpam-4396	151	1	j.	j.	PROPN
ejpam-4396	151	2	pure	pure	PROPN
ejpam-4396	151	3	appl	appl	PROPN
ejpam-4396	151	4	.	.	PROPN
ejpam-4396	151	5	math	math	PROPN
ejpam-4396	151	6	,	,	PUNCT
ejpam-4396	151	7	15	15	NUM
ejpam-4396	151	8	(	(	PUNCT
ejpam-4396	151	9	3	3	NUM
ejpam-4396	151	10	)	)	PUNCT
ejpam-4396	151	11	(	(	PUNCT
ejpam-4396	151	12	2022	2022	NUM
ejpam-4396	151	13	)	)	PUNCT
ejpam-4396	151	14	,	,	PUNCT
ejpam-4396	151	15	887	887	NUM
ejpam-4396	151	16	-	-	SYM
ejpam-4396	151	17	896	896	NUM
ejpam-4396	151	18	892	892	NUM
ejpam-4396	151	19	×6	×6	ADJ
ejpam-4396	151	20	1	1	NUM
ejpam-4396	151	21	5	5	NUM
ejpam-4396	151	22	1	1	NUM
ejpam-4396	151	23	1	1	NUM
ejpam-4396	151	24	5	5	NUM
ejpam-4396	151	25	5	5	NUM
ejpam-4396	151	26	5	5	NUM
ejpam-4396	151	27	1	1	NUM
ejpam-4396	151	28	table	table	NOUN
ejpam-4396	151	29	4	4	NUM
ejpam-4396	151	30	:	:	PUNCT
ejpam-4396	151	31	the	the	DET
ejpam-4396	151	32	g	g	PROPN
ejpam-4396	151	33	-	-	PUNCT
ejpam-4396	151	34	subgroup	subgroup	NOUN
ejpam-4396	151	35	h	h	NOUN
ejpam-4396	151	36	under	under	ADP
ejpam-4396	151	37	×6	×6	ADJ
ejpam-4396	151	38	×6	×6	ADJ
ejpam-4396	151	39	0	0	NUM
ejpam-4396	151	40	2	2	NUM
ejpam-4396	151	41	3	3	NUM
ejpam-4396	151	42	4	4	NUM
ejpam-4396	151	43	0	0	NUM
ejpam-4396	151	44	0	0	NUM
ejpam-4396	151	45	0	0	NUM
ejpam-4396	151	46	0	0	NUM
ejpam-4396	151	47	0	0	NUM
ejpam-4396	151	48	2	2	NUM
ejpam-4396	151	49	0	0	NUM
ejpam-4396	151	50	4	4	NUM
ejpam-4396	151	51	0	0	NUM
ejpam-4396	151	52	2	2	NUM
ejpam-4396	151	53	3	3	NUM
ejpam-4396	151	54	0	0	NUM
ejpam-4396	151	55	0	0	NUM
ejpam-4396	151	56	3	3	NUM
ejpam-4396	151	57	0	0	NUM
ejpam-4396	151	58	4	4	NUM
ejpam-4396	151	59	0	0	NUM
ejpam-4396	151	60	2	2	NUM
ejpam-4396	151	61	0	0	NUM
ejpam-4396	151	62	4	4	NUM
ejpam-4396	151	63	table	table	NOUN
ejpam-4396	151	64	5	5	NUM
ejpam-4396	151	65	:	:	PUNCT
ejpam-4396	151	66	the	the	DET
ejpam-4396	151	67	g	g	PROPN
ejpam-4396	151	68	-	-	PUNCT
ejpam-4396	151	69	subgroup	subgroup	NOUN
ejpam-4396	151	70	g\h	g\h	NOUN
ejpam-4396	151	71	=	=	PUNCT
ejpam-4396	151	72	{	{	PUNCT
ejpam-4396	151	73	0	0	NUM
ejpam-4396	151	74	,	,	PUNCT
ejpam-4396	151	75	2	2	NUM
ejpam-4396	151	76	,	,	PUNCT
ejpam-4396	151	77	3	3	NUM
ejpam-4396	151	78	,	,	PUNCT
ejpam-4396	151	79	4	4	NUM
ejpam-4396	151	80	}	}	PUNCT
ejpam-4396	151	81	under	under	ADP
ejpam-4396	151	82	×6	×6	ADJ
ejpam-4396	151	83	example	example	NOUN
ejpam-4396	151	84	1	1	NUM
ejpam-4396	151	85	.	.	PUNCT
ejpam-4396	151	86	consider	consider	VERB
ejpam-4396	151	87	the	the	DET
ejpam-4396	151	88	g	g	NOUN
ejpam-4396	151	89	-	-	PUNCT
ejpam-4396	151	90	group	group	NOUN
ejpam-4396	151	91	s6	s6	PROPN
ejpam-4396	151	92	=	=	SYM
ejpam-4396	151	93	{	{	PUNCT
ejpam-4396	151	94	0	0	NUM
ejpam-4396	151	95	,	,	PUNCT
ejpam-4396	151	96	1	1	NUM
ejpam-4396	151	97	,	,	PUNCT
ejpam-4396	151	98	2	2	NUM
ejpam-4396	151	99	,	,	PUNCT
ejpam-4396	151	100	3	3	NUM
ejpam-4396	151	101	,	,	PUNCT
ejpam-4396	151	102	4	4	NUM
ejpam-4396	151	103	,	,	PUNCT
ejpam-4396	151	104	5	5	NUM
ejpam-4396	151	105	}	}	PUNCT
ejpam-4396	151	106	under	under	ADP
ejpam-4396	151	107	multiplication	multiplication	NOUN
ejpam-4396	151	108	modulo	modulo	NOUN
ejpam-4396	151	109	6	6	NUM
ejpam-4396	151	110	.	.	PUNCT
ejpam-4396	151	111	note	note	VERB
ejpam-4396	151	112	that	that	SCONJ
ejpam-4396	151	113	the	the	DET
ejpam-4396	151	114	subsets	subset	NOUN
ejpam-4396	151	115	h	h	NOUN
ejpam-4396	152	1	=	=	PUNCT
ejpam-4396	152	2	{	{	PUNCT
ejpam-4396	152	3	1	1	NUM
ejpam-4396	152	4	,	,	PUNCT
ejpam-4396	152	5	5	5	NUM
ejpam-4396	152	6	}	}	PUNCT
ejpam-4396	152	7	and	and	CCONJ
ejpam-4396	152	8	g\h	g\h	PRON
ejpam-4396	152	9	=	=	PUNCT
ejpam-4396	152	10	{	{	PUNCT
ejpam-4396	152	11	0	0	NUM
ejpam-4396	152	12	,	,	PUNCT
ejpam-4396	152	13	2	2	NUM
ejpam-4396	152	14	,	,	PUNCT
ejpam-4396	152	15	3	3	NUM
ejpam-4396	152	16	,	,	PUNCT
ejpam-4396	152	17	4	4	NUM
ejpam-4396	152	18	}	}	PUNCT
ejpam-4396	152	19	are	be	AUX
ejpam-4396	152	20	g	g	NOUN
ejpam-4396	152	21	-	-	PUNCT
ejpam-4396	152	22	subgroups	subgroup	NOUN
ejpam-4396	152	23	of	of	ADP
ejpam-4396	152	24	(	(	PUNCT
ejpam-4396	152	25	s6,×6	s6,×6	PROPN
ejpam-4396	152	26	)	)	PUNCT
ejpam-4396	152	27	.	.	PUNCT
ejpam-4396	153	1	table	table	NOUN
ejpam-4396	153	2	4	4	NUM
ejpam-4396	153	3	and	and	CCONJ
ejpam-4396	153	4	table	table	NOUN
ejpam-4396	153	5	5	5	NUM
ejpam-4396	153	6	may	may	AUX
ejpam-4396	153	7	be	be	AUX
ejpam-4396	153	8	helpful	helpful	ADJ
ejpam-4396	153	9	in	in	ADP
ejpam-4396	153	10	seeing	see	VERB
ejpam-4396	153	11	this	this	PRON
ejpam-4396	153	12	.	.	PUNCT
ejpam-4396	154	1	if	if	SCONJ
ejpam-4396	154	2	e	e	PROPN
ejpam-4396	154	3	is	be	AUX
ejpam-4396	154	4	an	an	DET
ejpam-4396	154	5	identity	identity	NOUN
ejpam-4396	154	6	of	of	ADP
ejpam-4396	154	7	an	an	DET
ejpam-4396	154	8	element	element	NOUN
ejpam-4396	154	9	in	in	ADP
ejpam-4396	154	10	g	g	PROPN
ejpam-4396	154	11	,	,	PUNCT
ejpam-4396	154	12	say	say	VERB
ejpam-4396	154	13	x	x	NOUN
ejpam-4396	154	14	,	,	PUNCT
ejpam-4396	154	15	such	such	ADJ
ejpam-4396	154	16	that	that	SCONJ
ejpam-4396	154	17	there	there	PRON
ejpam-4396	154	18	an	an	DET
ejpam-4396	154	19	element	element	NOUN
ejpam-4396	154	20	y	y	PROPN
ejpam-4396	154	21	with	with	ADP
ejpam-4396	154	22	xy	xy	PROPN
ejpam-4396	155	1	=	=	SYM
ejpam-4396	155	2	e	e	NOUN
ejpam-4396	155	3	,	,	PUNCT
ejpam-4396	155	4	then	then	ADV
ejpam-4396	155	5	we	we	PRON
ejpam-4396	155	6	say	say	VERB
ejpam-4396	155	7	that	that	SCONJ
ejpam-4396	155	8	e	e	NOUN
ejpam-4396	155	9	is	be	AUX
ejpam-4396	155	10	a	a	DET
ejpam-4396	155	11	leaf	leaf	NOUN
ejpam-4396	155	12	.	.	PUNCT
ejpam-4396	156	1	the	the	DET
ejpam-4396	156	2	next	next	ADJ
ejpam-4396	156	3	theorem	theorem	NOUN
ejpam-4396	156	4	,	,	PUNCT
ejpam-4396	156	5	theorem	theorem	VERB
ejpam-4396	156	6	3	3	NUM
ejpam-4396	156	7	,	,	PUNCT
ejpam-4396	156	8	provides	provide	VERB
ejpam-4396	156	9	another	another	DET
ejpam-4396	156	10	way	way	NOUN
ejpam-4396	156	11	of	of	ADP
ejpam-4396	156	12	constructing	construct	VERB
ejpam-4396	156	13	a	a	DET
ejpam-4396	156	14	g	g	NOUN
ejpam-4396	156	15	-	-	PUNCT
ejpam-4396	156	16	subgroup	subgroup	NOUN
ejpam-4396	156	17	.	.	PUNCT
ejpam-4396	157	1	it	it	PRON
ejpam-4396	157	2	says	say	VERB
ejpam-4396	157	3	that	that	SCONJ
ejpam-4396	157	4	the	the	DET
ejpam-4396	157	5	set	set	NOUN
ejpam-4396	157	6	of	of	ADP
ejpam-4396	157	7	all	all	DET
ejpam-4396	157	8	leaf	leaf	NOUN
ejpam-4396	157	9	in	in	ADP
ejpam-4396	157	10	a	a	DET
ejpam-4396	157	11	g	g	NOUN
ejpam-4396	157	12	-	-	PUNCT
ejpam-4396	157	13	group	group	NOUN
ejpam-4396	157	14	is	be	AUX
ejpam-4396	157	15	a	a	DET
ejpam-4396	157	16	g	g	NOUN
ejpam-4396	157	17	-	-	PUNCT
ejpam-4396	157	18	subgroup	subgroup	NOUN
ejpam-4396	157	19	.	.	PUNCT
ejpam-4396	158	1	theorem	theorem	NOUN
ejpam-4396	158	2	3	3	X
ejpam-4396	158	3	.	.	PUNCT
ejpam-4396	159	1	let	let	VERB
ejpam-4396	159	2	g	g	NOUN
ejpam-4396	159	3	be	be	AUX
ejpam-4396	159	4	an	an	DET
ejpam-4396	159	5	abelian	abelian	ADJ
ejpam-4396	159	6	g	g	NOUN
ejpam-4396	159	7	-	-	PUNCT
ejpam-4396	159	8	group	group	NOUN
ejpam-4396	159	9	.	.	PUNCT
ejpam-4396	160	1	if	if	SCONJ
ejpam-4396	160	2	e	e	X
ejpam-4396	160	3	=	=	PRON
ejpam-4396	160	4	{	{	PUNCT
ejpam-4396	160	5	e	e	PROPN
ejpam-4396	160	6	∈	∈	PROPN
ejpam-4396	160	7	g	g	NOUN
ejpam-4396	160	8	:	:	PUNCT
ejpam-4396	160	9	e	e	X
ejpam-4396	160	10	is	be	AUX
ejpam-4396	160	11	a	a	DET
ejpam-4396	160	12	leaf	leaf	NOUN
ejpam-4396	160	13	}	}	PUNCT
ejpam-4396	160	14	,	,	PUNCT
ejpam-4396	160	15	then	then	ADV
ejpam-4396	160	16	e	e	PROPN
ejpam-4396	160	17	is	be	AUX
ejpam-4396	160	18	a	a	DET
ejpam-4396	160	19	g	g	NOUN
ejpam-4396	160	20	-	-	PUNCT
ejpam-4396	160	21	subgroup	subgroup	NOUN
ejpam-4396	160	22	of	of	ADP
ejpam-4396	160	23	g.	g.	PROPN
ejpam-4396	160	24	proof	proof	PROPN
ejpam-4396	160	25	.	.	PUNCT
ejpam-4396	161	1	(	(	PUNCT
ejpam-4396	161	2	g1	g1	PROPN
ejpam-4396	161	3	)	)	PUNCT
ejpam-4396	161	4	follows	follow	VERB
ejpam-4396	161	5	from	from	ADP
ejpam-4396	161	6	the	the	DET
ejpam-4396	161	7	fact	fact	NOUN
ejpam-4396	161	8	that	that	SCONJ
ejpam-4396	161	9	e	e	NOUN
ejpam-4396	161	10	is	be	AUX
ejpam-4396	161	11	a	a	DET
ejpam-4396	161	12	subset	subset	NOUN
ejpam-4396	161	13	of	of	ADP
ejpam-4396	161	14	g.	g.	PROPN
ejpam-4396	161	15	next	next	ADV
ejpam-4396	161	16	,	,	PUNCT
ejpam-4396	161	17	since	since	SCONJ
ejpam-4396	161	18	each	each	DET
ejpam-4396	161	19	element	element	NOUN
ejpam-4396	161	20	,	,	PUNCT
ejpam-4396	161	21	say	say	VERB
ejpam-4396	161	22	e	e	NOUN
ejpam-4396	161	23	,	,	PUNCT
ejpam-4396	161	24	of	of	ADP
ejpam-4396	161	25	e	e	PROPN
ejpam-4396	161	26	is	be	AUX
ejpam-4396	161	27	a	a	DET
ejpam-4396	161	28	leaf	leaf	NOUN
ejpam-4396	161	29	,	,	PUNCT
ejpam-4396	161	30	it	it	PRON
ejpam-4396	161	31	is	be	AUX
ejpam-4396	161	32	an	an	DET
ejpam-4396	161	33	identity	identity	NOUN
ejpam-4396	161	34	of	of	ADP
ejpam-4396	161	35	some	some	DET
ejpam-4396	161	36	element	element	NOUN
ejpam-4396	161	37	,	,	PUNCT
ejpam-4396	161	38	say	say	VERB
ejpam-4396	161	39	x	x	X
ejpam-4396	161	40	,	,	PUNCT
ejpam-4396	161	41	with	with	ADP
ejpam-4396	161	42	a	a	DET
ejpam-4396	161	43	property	property	NOUN
ejpam-4396	161	44	that	that	PRON
ejpam-4396	161	45	there	there	PRON
ejpam-4396	161	46	exist	exist	VERB
ejpam-4396	161	47	y	y	PROPN
ejpam-4396	161	48	∈	∈	PROPN
ejpam-4396	161	49	g	g	PROPN
ejpam-4396	161	50	with	with	ADP
ejpam-4396	161	51	xy	xy	PROPN
ejpam-4396	161	52	=	=	PUNCT
ejpam-4396	161	53	e.	e.	PROPN
ejpam-4396	161	54	since	since	SCONJ
ejpam-4396	161	55	ex	ex	X
ejpam-4396	161	56	=	=	SYM
ejpam-4396	161	57	x	x	NOUN
ejpam-4396	161	58	,	,	PUNCT
ejpam-4396	161	59	we	we	PRON
ejpam-4396	161	60	have	have	VERB
ejpam-4396	161	61	exy	exy	ADJ
ejpam-4396	162	1	=	=	SYM
ejpam-4396	162	2	xy	xy	PROPN
ejpam-4396	162	3	.	.	PUNCT
ejpam-4396	163	1	and	and	CCONJ
ejpam-4396	163	2	so	so	ADV
ejpam-4396	163	3	,	,	PUNCT
ejpam-4396	163	4	ee	ee	PROPN
ejpam-4396	163	5	=	=	PROPN
ejpam-4396	163	6	e.	e.	PROPN
ejpam-4396	163	7	hence	hence	ADV
ejpam-4396	163	8	,	,	PUNCT
ejpam-4396	163	9	e	e	PROPN
ejpam-4396	163	10	has	have	VERB
ejpam-4396	163	11	an	an	DET
ejpam-4396	163	12	identity	identity	NOUN
ejpam-4396	163	13	(	(	PUNCT
ejpam-4396	163	14	which	which	PRON
ejpam-4396	163	15	is	be	AUX
ejpam-4396	163	16	itself	itself	PRON
ejpam-4396	163	17	)	)	PUNCT
ejpam-4396	163	18	and	and	CCONJ
ejpam-4396	163	19	an	an	DET
ejpam-4396	163	20	inverse	inverse	NOUN
ejpam-4396	163	21	(	(	PUNCT
ejpam-4396	163	22	which	which	PRON
ejpam-4396	163	23	is	be	AUX
ejpam-4396	163	24	itself	itself	PRON
ejpam-4396	163	25	also	also	ADV
ejpam-4396	163	26	)	)	PUNCT
ejpam-4396	163	27	.	.	PUNCT
ejpam-4396	164	1	this	this	PRON
ejpam-4396	164	2	shows	show	VERB
ejpam-4396	164	3	that	that	SCONJ
ejpam-4396	164	4	e	e	NOUN
ejpam-4396	164	5	satisfies	satisfie	NOUN
ejpam-4396	164	6	(	(	PUNCT
ejpam-4396	164	7	g2	g2	PROPN
ejpam-4396	164	8	)	)	PUNCT
ejpam-4396	164	9	and	and	CCONJ
ejpam-4396	164	10	(	(	PUNCT
ejpam-4396	164	11	g3	g3	PROPN
ejpam-4396	164	12	)	)	PUNCT
ejpam-4396	164	13	.	.	PUNCT
ejpam-4396	165	1	accordingly	accordingly	ADV
ejpam-4396	165	2	,	,	PUNCT
ejpam-4396	165	3	e	e	X
ejpam-4396	165	4	is	be	AUX
ejpam-4396	165	5	a	a	DET
ejpam-4396	165	6	g	g	NOUN
ejpam-4396	165	7	-	-	PUNCT
ejpam-4396	165	8	subgroup	subgroup	NOUN
ejpam-4396	165	9	of	of	ADP
ejpam-4396	165	10	g.	g.	PROPN
ejpam-4396	165	11	this	this	DET
ejpam-4396	165	12	section	section	NOUN
ejpam-4396	165	13	is	be	AUX
ejpam-4396	165	14	culminated	culminate	VERB
ejpam-4396	165	15	with	with	ADP
ejpam-4396	165	16	two	two	NUM
ejpam-4396	165	17	obvious	obvious	ADJ
ejpam-4396	165	18	remarks	remark	NOUN
ejpam-4396	165	19	,	,	PUNCT
ejpam-4396	165	20	remark	remark	VERB
ejpam-4396	165	21	6	6	NUM
ejpam-4396	165	22	and	and	CCONJ
ejpam-4396	165	23	remark	remark	VERB
ejpam-4396	165	24	7	7	NUM
ejpam-4396	165	25	.	.	NOUN
ejpam-4396	165	26	remark	remark	NOUN
ejpam-4396	165	27	6	6	NUM
ejpam-4396	165	28	is	be	AUX
ejpam-4396	165	29	evident	evident	ADJ
ejpam-4396	165	30	from	from	ADP
ejpam-4396	165	31	example	example	NOUN
ejpam-4396	165	32	1	1	NUM
ejpam-4396	165	33	.	.	PUNCT
ejpam-4396	166	1	this	this	PRON
ejpam-4396	166	2	is	be	AUX
ejpam-4396	166	3	not	not	PART
ejpam-4396	166	4	always	always	ADV
ejpam-4396	166	5	the	the	DET
ejpam-4396	166	6	case	case	NOUN
ejpam-4396	166	7	for	for	ADP
ejpam-4396	166	8	other	other	ADJ
ejpam-4396	166	9	structures	structure	NOUN
ejpam-4396	166	10	.	.	PUNCT
ejpam-4396	167	1	remark	remark	NOUN
ejpam-4396	167	2	6	6	NUM
ejpam-4396	167	3	.	.	PUNCT
ejpam-4396	168	1	the	the	DET
ejpam-4396	168	2	intersection	intersection	NOUN
ejpam-4396	168	3	of	of	ADP
ejpam-4396	168	4	g	g	NOUN
ejpam-4396	168	5	-	-	PUNCT
ejpam-4396	168	6	subgroups	subgroup	NOUN
ejpam-4396	168	7	may	may	AUX
ejpam-4396	168	8	not	not	PART
ejpam-4396	168	9	be	be	AUX
ejpam-4396	168	10	a	a	DET
ejpam-4396	168	11	g	g	NOUN
ejpam-4396	168	12	-	-	PUNCT
ejpam-4396	168	13	subgroup	subgroup	NOUN
ejpam-4396	168	14	.	.	PUNCT
ejpam-4396	169	1	remark	remark	PROPN
ejpam-4396	169	2	7	7	NUM
ejpam-4396	169	3	.	.	PUNCT
ejpam-4396	170	1	the	the	DET
ejpam-4396	170	2	union	union	NOUN
ejpam-4396	170	3	of	of	ADP
ejpam-4396	170	4	g	g	NOUN
ejpam-4396	170	5	-	-	PUNCT
ejpam-4396	170	6	subgroups	subgroup	NOUN
ejpam-4396	170	7	is	be	AUX
ejpam-4396	170	8	not	not	PART
ejpam-4396	170	9	always	always	ADV
ejpam-4396	170	10	a	a	DET
ejpam-4396	170	11	g	g	NOUN
ejpam-4396	170	12	-	-	PUNCT
ejpam-4396	170	13	subgroup	subgroup	NOUN
ejpam-4396	170	14	.	.	PUNCT
ejpam-4396	171	1	to	to	PART
ejpam-4396	171	2	see	see	VERB
ejpam-4396	171	3	this	this	PRON
ejpam-4396	171	4	,	,	PUNCT
ejpam-4396	171	5	we	we	PRON
ejpam-4396	171	6	note	note	VERB
ejpam-4396	171	7	that	that	SCONJ
ejpam-4396	171	8	h1	h1	PROPN
ejpam-4396	171	9	=	=	SYM
ejpam-4396	171	10	{	{	PUNCT
ejpam-4396	171	11	2	2	NUM
ejpam-4396	171	12	,	,	PUNCT
ejpam-4396	171	13	4	4	NUM
ejpam-4396	171	14	}	}	PUNCT
ejpam-4396	171	15	and	and	CCONJ
ejpam-4396	171	16	h2	h2	NOUN
ejpam-4396	171	17	=	=	SYM
ejpam-4396	171	18	{	{	PUNCT
ejpam-4396	171	19	1	1	NUM
ejpam-4396	171	20	,	,	PUNCT
ejpam-4396	171	21	3	3	NUM
ejpam-4396	171	22	}	}	PUNCT
ejpam-4396	171	23	are	be	AUX
ejpam-4396	171	24	g	g	NOUN
ejpam-4396	171	25	-	-	PUNCT
ejpam-4396	171	26	subgroups	subgroup	NOUN
ejpam-4396	171	27	of	of	ADP
ejpam-4396	171	28	s6	s6	PROPN
ejpam-4396	171	29	=	=	SYM
ejpam-4396	171	30	(	(	PUNCT
ejpam-4396	171	31	{	{	PUNCT
ejpam-4396	171	32	0	0	NUM
ejpam-4396	171	33	,	,	PUNCT
ejpam-4396	171	34	1	1	NUM
ejpam-4396	171	35	,	,	PUNCT
ejpam-4396	171	36	2	2	NUM
ejpam-4396	171	37	,	,	PUNCT
ejpam-4396	171	38	3	3	NUM
ejpam-4396	171	39	,	,	PUNCT
ejpam-4396	171	40	4	4	NUM
ejpam-4396	171	41	,	,	PUNCT
ejpam-4396	171	42	5},×6	5},×6	NUM
ejpam-4396	171	43	)	)	PUNCT
ejpam-4396	171	44	,	,	PUNCT
ejpam-4396	171	45	but	but	CCONJ
ejpam-4396	171	46	their	their	PRON
ejpam-4396	171	47	union	union	NOUN
ejpam-4396	171	48	h1	h1	VERB
ejpam-4396	171	49	∪h2	∪h2	PROPN
ejpam-4396	171	50	is	be	AUX
ejpam-4396	171	51	not	not	PART
ejpam-4396	171	52	a	a	DET
ejpam-4396	171	53	g	g	NOUN
ejpam-4396	171	54	-	-	PUNCT
ejpam-4396	171	55	subgroup	subgroup	NOUN
ejpam-4396	171	56	.	.	PUNCT
ejpam-4396	172	1	4	4	X
ejpam-4396	172	2	.	.	X
ejpam-4396	172	3	homomorphism	homomorphism	NOUN
ejpam-4396	172	4	in	in	ADP
ejpam-4396	172	5	this	this	DET
ejpam-4396	172	6	section	section	NOUN
ejpam-4396	172	7	,	,	PUNCT
ejpam-4396	172	8	we	we	PRON
ejpam-4396	172	9	present	present	VERB
ejpam-4396	172	10	some	some	DET
ejpam-4396	172	11	[	[	X
ejpam-4396	172	12	homomorphism	homomorphism	X
ejpam-4396	172	13	]	]	PUNCT
ejpam-4396	172	14	conditions	condition	NOUN
ejpam-4396	172	15	that	that	SCONJ
ejpam-4396	172	16	we	we	PRON
ejpam-4396	172	17	will	will	AUX
ejpam-4396	172	18	impose	impose	VERB
ejpam-4396	172	19	on	on	ADP
ejpam-4396	172	20	a	a	DET
ejpam-4396	172	21	function	function	NOUN
ejpam-4396	172	22	so	so	SCONJ
ejpam-4396	172	23	that	that	SCONJ
ejpam-4396	172	24	it	it	PRON
ejpam-4396	172	25	will	will	AUX
ejpam-4396	172	26	be	be	AUX
ejpam-4396	172	27	able	able	ADJ
ejpam-4396	172	28	to	to	PART
ejpam-4396	172	29	identify	identify	VERB
ejpam-4396	172	30	the	the	DET
ejpam-4396	172	31	units	unit	NOUN
ejpam-4396	172	32	of	of	ADP
ejpam-4396	172	33	one	one	NUM
ejpam-4396	172	34	g	g	NOUN
ejpam-4396	172	35	-	-	PUNCT
ejpam-4396	172	36	group	group	NOUN
ejpam-4396	172	37	with	with	ADP
ejpam-4396	172	38	the	the	DET
ejpam-4396	172	39	units	unit	NOUN
ejpam-4396	172	40	of	of	ADP
ejpam-4396	172	41	another	another	PRON
ejpam-4396	172	42	.	.	PUNCT
ejpam-4396	173	1	the	the	DET
ejpam-4396	173	2	main	main	ADJ
ejpam-4396	173	3	objective	objective	NOUN
ejpam-4396	173	4	here	here	ADV
ejpam-4396	173	5	is	be	AUX
ejpam-4396	173	6	to	to	PART
ejpam-4396	173	7	provide	provide	VERB
ejpam-4396	173	8	another	another	DET
ejpam-4396	173	9	way	way	NOUN
ejpam-4396	173	10	of	of	ADP
ejpam-4396	173	11	constructing	construct	VERB
ejpam-4396	173	12	a	a	DET
ejpam-4396	173	13	g	g	NOUN
ejpam-4396	173	14	-	-	PUNCT
ejpam-4396	173	15	subgroup	subgroup	NOUN
ejpam-4396	173	16	.	.	PUNCT
ejpam-4396	174	1	the	the	DET
ejpam-4396	174	2	next	next	ADJ
ejpam-4396	174	3	statement	statement	NOUN
ejpam-4396	174	4	,	,	PUNCT
ejpam-4396	174	5	definition	definition	NOUN
ejpam-4396	174	6	2	2	NUM
ejpam-4396	174	7	,	,	PUNCT
ejpam-4396	174	8	describes	describe	VERB
ejpam-4396	174	9	what	what	PRON
ejpam-4396	174	10	a	a	DET
ejpam-4396	174	11	homomorphism	homomorphism	NOUN
ejpam-4396	174	12	is	be	AUX
ejpam-4396	174	13	.	.	PUNCT
ejpam-4396	175	1	j.	j.	PROPN
ejpam-4396	175	2	caraquil	caraquil	PROPN
ejpam-4396	175	3	,	,	PUNCT
ejpam-4396	175	4	m.	m.	PROPN
ejpam-4396	175	5	baldado	baldado	PROPN
ejpam-4396	175	6	jr	jr	PROPN
ejpam-4396	175	7	.	.	PROPN
ejpam-4396	175	8	/	/	SYM
ejpam-4396	175	9	eur	eur	PROPN
ejpam-4396	175	10	.	.	PUNCT
ejpam-4396	176	1	j.	j.	PROPN
ejpam-4396	176	2	pure	pure	PROPN
ejpam-4396	176	3	appl	appl	PROPN
ejpam-4396	176	4	.	.	PROPN
ejpam-4396	176	5	math	math	PROPN
ejpam-4396	176	6	,	,	PUNCT
ejpam-4396	176	7	15	15	NUM
ejpam-4396	176	8	(	(	PUNCT
ejpam-4396	176	9	3	3	NUM
ejpam-4396	176	10	)	)	PUNCT
ejpam-4396	176	11	(	(	PUNCT
ejpam-4396	176	12	2022	2022	NUM
ejpam-4396	176	13	)	)	PUNCT
ejpam-4396	176	14	,	,	PUNCT
ejpam-4396	176	15	887	887	NUM
ejpam-4396	176	16	-	-	SYM
ejpam-4396	176	17	896	896	NUM
ejpam-4396	176	18	893	893	NUM
ejpam-4396	176	19	definition	definition	NOUN
ejpam-4396	176	20	2	2	NUM
ejpam-4396	176	21	.	.	PUNCT
ejpam-4396	177	1	let	let	VERB
ejpam-4396	177	2	g	g	PROPN
ejpam-4396	177	3	and	and	CCONJ
ejpam-4396	177	4	j	j	PROPN
ejpam-4396	177	5	be	be	VERB
ejpam-4396	177	6	g	g	NOUN
ejpam-4396	177	7	-	-	PUNCT
ejpam-4396	177	8	groups	group	NOUN
ejpam-4396	177	9	with	with	ADP
ejpam-4396	177	10	binary	binary	ADJ
ejpam-4396	177	11	operations	operation	NOUN
ejpam-4396	177	12	∗	∗	NOUN
ejpam-4396	177	13	and	and	CCONJ
ejpam-4396	177	14	∗′	∗′	PROPN
ejpam-4396	177	15	,	,	PUNCT
ejpam-4396	177	16	respectively	respectively	ADV
ejpam-4396	177	17	.	.	PUNCT
ejpam-4396	178	1	a	a	DET
ejpam-4396	178	2	function	function	NOUN
ejpam-4396	178	3	f	f	NOUN
ejpam-4396	178	4	:	:	PUNCT
ejpam-4396	178	5	g	g	PROPN
ejpam-4396	178	6	→	→	SYM
ejpam-4396	178	7	j	j	PROPN
ejpam-4396	178	8	is	be	AUX
ejpam-4396	178	9	a	a	DET
ejpam-4396	178	10	homomorphism	homomorphism	NOUN
ejpam-4396	178	11	if	if	SCONJ
ejpam-4396	178	12	f(a	f(a	PROPN
ejpam-4396	178	13	∗	∗	X
ejpam-4396	178	14	b	b	NOUN
ejpam-4396	178	15	)	)	PUNCT
ejpam-4396	178	16	=	=	SYM
ejpam-4396	178	17	f(a	f(a	NOUN
ejpam-4396	178	18	)	)	PUNCT
ejpam-4396	178	19	∗′	∗′	ADJ
ejpam-4396	178	20	f(b	f(b	NOUN
ejpam-4396	178	21	)	)	PUNCT
ejpam-4396	178	22	.	.	PUNCT
ejpam-4396	179	1	example	example	NOUN
ejpam-4396	180	1	2	2	X
ejpam-4396	180	2	.	.	X
ejpam-4396	180	3	consider	consider	VERB
ejpam-4396	180	4	again	again	ADV
ejpam-4396	180	5	the	the	DET
ejpam-4396	180	6	g	g	PROPN
ejpam-4396	180	7	-	-	PUNCT
ejpam-4396	180	8	group	group	NOUN
ejpam-4396	180	9	s6	s6	PROPN
ejpam-4396	180	10	=	=	SYM
ejpam-4396	180	11	{	{	PUNCT
ejpam-4396	180	12	0	0	NUM
ejpam-4396	180	13	,	,	PUNCT
ejpam-4396	180	14	1	1	NUM
ejpam-4396	180	15	,	,	PUNCT
ejpam-4396	180	16	2	2	NUM
ejpam-4396	180	17	,	,	PUNCT
ejpam-4396	180	18	3	3	NUM
ejpam-4396	180	19	,	,	PUNCT
ejpam-4396	180	20	4	4	NUM
ejpam-4396	180	21	,	,	PUNCT
ejpam-4396	180	22	5	5	NUM
ejpam-4396	180	23	}	}	PUNCT
ejpam-4396	180	24	under	under	ADP
ejpam-4396	180	25	multiplication	multiplication	NOUN
ejpam-4396	180	26	modulo	modulo	NOUN
ejpam-4396	180	27	6	6	NUM
ejpam-4396	180	28	.	.	PUNCT
ejpam-4396	181	1	in	in	ADP
ejpam-4396	181	2	example	example	NOUN
ejpam-4396	181	3	1	1	NUM
ejpam-4396	181	4	,	,	PUNCT
ejpam-4396	181	5	the	the	DET
ejpam-4396	181	6	subsets	subset	NOUN
ejpam-4396	181	7	h	h	NOUN
ejpam-4396	181	8	=	=	PUNCT
ejpam-4396	181	9	{	{	PUNCT
ejpam-4396	181	10	1	1	NUM
ejpam-4396	181	11	,	,	PUNCT
ejpam-4396	181	12	5	5	NUM
ejpam-4396	181	13	}	}	PUNCT
ejpam-4396	181	14	and	and	CCONJ
ejpam-4396	181	15	g\h	g\h	PRON
ejpam-4396	181	16	=	=	PUNCT
ejpam-4396	181	17	{	{	PUNCT
ejpam-4396	181	18	0	0	NUM
ejpam-4396	181	19	,	,	PUNCT
ejpam-4396	181	20	2	2	NUM
ejpam-4396	181	21	,	,	PUNCT
ejpam-4396	181	22	3	3	NUM
ejpam-4396	181	23	,	,	PUNCT
ejpam-4396	181	24	4	4	NUM
ejpam-4396	181	25	}	}	PUNCT
ejpam-4396	181	26	are	be	AUX
ejpam-4396	181	27	both	both	PRON
ejpam-4396	181	28	g	g	NOUN
ejpam-4396	181	29	-	-	PUNCT
ejpam-4396	181	30	subgroups	subgroup	NOUN
ejpam-4396	181	31	of	of	ADP
ejpam-4396	181	32	s6	s6	PROPN
ejpam-4396	181	33	.	.	PUNCT
ejpam-4396	182	1	now	now	ADV
ejpam-4396	182	2	,	,	PUNCT
ejpam-4396	182	3	define	define	VERB
ejpam-4396	182	4	f	f	X
ejpam-4396	182	5	:	:	PUNCT
ejpam-4396	182	6	h	h	PROPN
ejpam-4396	182	7	→	→	PUNCT
ejpam-4396	182	8	g\h	g\h	PROPN
ejpam-4396	182	9	by	by	ADP
ejpam-4396	182	10	1	1	NUM
ejpam-4396	182	11	7→	7→	NUM
ejpam-4396	182	12	4	4	NUM
ejpam-4396	182	13	and	and	CCONJ
ejpam-4396	182	14	5	5	NUM
ejpam-4396	182	15	7→	7→	NUM
ejpam-4396	182	16	2	2	NUM
ejpam-4396	182	17	.	.	PUNCT
ejpam-4396	183	1	then	then	ADV
ejpam-4396	183	2	it	it	PRON
ejpam-4396	183	3	is	be	AUX
ejpam-4396	183	4	easy	easy	ADJ
ejpam-4396	183	5	to	to	PART
ejpam-4396	183	6	see	see	VERB
ejpam-4396	183	7	that	that	SCONJ
ejpam-4396	183	8	f	f	PROPN
ejpam-4396	183	9	is	be	AUX
ejpam-4396	183	10	a	a	DET
ejpam-4396	183	11	homomorphism	homomorphism	NOUN
ejpam-4396	183	12	.	.	PUNCT
ejpam-4396	184	1	let	let	VERB
ejpam-4396	184	2	g	g	PRON
ejpam-4396	184	3	be	be	AUX
ejpam-4396	184	4	an	an	DET
ejpam-4396	184	5	abelian	abelian	ADJ
ejpam-4396	184	6	g	g	NOUN
ejpam-4396	184	7	-	-	PUNCT
ejpam-4396	184	8	group	group	NOUN
ejpam-4396	184	9	.	.	PUNCT
ejpam-4396	185	1	since	since	SCONJ
ejpam-4396	185	2	by	by	ADP
ejpam-4396	185	3	corollary	corollary	ADJ
ejpam-4396	185	4	1	1	NUM
ejpam-4396	185	5	,	,	PUNCT
ejpam-4396	185	6	h	h	NOUN
ejpam-4396	185	7	=	=	PRON
ejpam-4396	185	8	{	{	PUNCT
ejpam-4396	185	9	h	h	NOUN
ejpam-4396	185	10	∈	∈	PROPN
ejpam-4396	185	11	g	g	PROPN
ejpam-4396	185	12	:	:	PUNCT
ejpam-4396	185	13	h	h	PROPN
ejpam-4396	185	14	is	be	AUX
ejpam-4396	185	15	a	a	DET
ejpam-4396	185	16	unit	unit	NOUN
ejpam-4396	185	17	}	}	PUNCT
ejpam-4396	185	18	is	be	AUX
ejpam-4396	185	19	a	a	DET
ejpam-4396	185	20	group	group	NOUN
ejpam-4396	185	21	,	,	PUNCT
ejpam-4396	185	22	all	all	DET
ejpam-4396	185	23	the	the	DET
ejpam-4396	185	24	established	establish	VERB
ejpam-4396	185	25	properties	property	NOUN
ejpam-4396	185	26	of	of	ADP
ejpam-4396	185	27	a	a	DET
ejpam-4396	185	28	homomorphism	homomorphism	NOUN
ejpam-4396	185	29	f	f	X
ejpam-4396	185	30	:	:	PUNCT
ejpam-4396	185	31	h	h	NOUN
ejpam-4396	185	32	→	→	SYM
ejpam-4396	185	33	h	h	NOUN
ejpam-4396	185	34	for	for	ADP
ejpam-4396	185	35	groups	group	NOUN
ejpam-4396	185	36	should	should	AUX
ejpam-4396	185	37	hold	hold	VERB
ejpam-4396	185	38	.	.	PUNCT
ejpam-4396	186	1	the	the	DET
ejpam-4396	186	2	next	next	ADJ
ejpam-4396	186	3	theorem	theorem	NOUN
ejpam-4396	186	4	,	,	PUNCT
ejpam-4396	186	5	theorem	theorem	VERB
ejpam-4396	186	6	4	4	NUM
ejpam-4396	186	7	,	,	PUNCT
ejpam-4396	186	8	say	say	VERB
ejpam-4396	186	9	that	that	SCONJ
ejpam-4396	186	10	under	under	ADP
ejpam-4396	186	11	a	a	DET
ejpam-4396	186	12	homomorphism	homomorphism	NOUN
ejpam-4396	186	13	the	the	DET
ejpam-4396	186	14	image	image	NOUN
ejpam-4396	186	15	of	of	ADP
ejpam-4396	186	16	an	an	DET
ejpam-4396	186	17	identity	identity	NOUN
ejpam-4396	186	18	is	be	AUX
ejpam-4396	186	19	an	an	DET
ejpam-4396	186	20	identity	identity	NOUN
ejpam-4396	186	21	in	in	ADP
ejpam-4396	186	22	the	the	DET
ejpam-4396	186	23	co	co	NOUN
ejpam-4396	186	24	-	-	NOUN
ejpam-4396	186	25	domain	domain	NOUN
ejpam-4396	186	26	,	,	PUNCT
ejpam-4396	186	27	and	and	CCONJ
ejpam-4396	186	28	the	the	DET
ejpam-4396	186	29	image	image	NOUN
ejpam-4396	186	30	of	of	ADP
ejpam-4396	186	31	an	an	DET
ejpam-4396	186	32	inverse	inverse	NOUN
ejpam-4396	186	33	is	be	AUX
ejpam-4396	186	34	an	an	DET
ejpam-4396	186	35	inverse	inverse	NOUN
ejpam-4396	186	36	in	in	ADP
ejpam-4396	186	37	the	the	DET
ejpam-4396	186	38	co	co	NOUN
ejpam-4396	186	39	-	-	NOUN
ejpam-4396	186	40	domain	domain	NOUN
ejpam-4396	186	41	.	.	PUNCT
ejpam-4396	187	1	theorem	theorem	NOUN
ejpam-4396	187	2	4	4	NUM
ejpam-4396	187	3	.	.	PUNCT
ejpam-4396	188	1	let	let	VERB
ejpam-4396	188	2	g	g	PROPN
ejpam-4396	188	3	and	and	CCONJ
ejpam-4396	188	4	j	j	PROPN
ejpam-4396	188	5	be	be	VERB
ejpam-4396	188	6	abelian	abelian	ADJ
ejpam-4396	188	7	g	g	NOUN
ejpam-4396	188	8	-	-	PUNCT
ejpam-4396	188	9	groups	group	NOUN
ejpam-4396	188	10	and	and	CCONJ
ejpam-4396	188	11	a	a	DET
ejpam-4396	188	12	∈	∈	NOUN
ejpam-4396	188	13	g	g	NOUN
ejpam-4396	188	14	with	with	ADP
ejpam-4396	188	15	identity	identity	NOUN
ejpam-4396	188	16	e	e	NOUN
ejpam-4396	188	17	and	and	CCONJ
ejpam-4396	188	18	inverse	inverse	PROPN
ejpam-4396	188	19	b.	b.	PROPN
ejpam-4396	189	1	if	if	SCONJ
ejpam-4396	189	2	f	f	PROPN
ejpam-4396	189	3	:	:	PUNCT
ejpam-4396	189	4	g	g	PROPN
ejpam-4396	189	5	→	→	SYM
ejpam-4396	189	6	j	j	PROPN
ejpam-4396	189	7	is	be	AUX
ejpam-4396	189	8	a	a	DET
ejpam-4396	189	9	homomorphism	homomorphism	NOUN
ejpam-4396	189	10	,	,	PUNCT
ejpam-4396	189	11	then	then	ADV
ejpam-4396	189	12	a.	a.	NOUN
ejpam-4396	189	13	)	)	PUNCT
ejpam-4396	189	14	f(e	f(e	NOUN
ejpam-4396	189	15	)	)	PUNCT
ejpam-4396	189	16	is	be	AUX
ejpam-4396	189	17	an	an	DET
ejpam-4396	189	18	identity	identity	NOUN
ejpam-4396	189	19	of	of	ADP
ejpam-4396	189	20	f(a	f(a	NOUN
ejpam-4396	189	21	)	)	PUNCT
ejpam-4396	189	22	,	,	PUNCT
ejpam-4396	189	23	and	and	CCONJ
ejpam-4396	189	24	b.	b.	PROPN
ejpam-4396	189	25	)	)	PUNCT
ejpam-4396	189	26	f(b	f(b	PROPN
ejpam-4396	189	27	)	)	PUNCT
ejpam-4396	189	28	is	be	AUX
ejpam-4396	189	29	an	an	DET
ejpam-4396	189	30	inverse	inverse	NOUN
ejpam-4396	189	31	of	of	ADP
ejpam-4396	189	32	f(a	f(a	PROPN
ejpam-4396	189	33	)	)	PUNCT
ejpam-4396	189	34	.	.	PUNCT
ejpam-4396	190	1	proof	proof	NOUN
ejpam-4396	190	2	.	.	PUNCT
ejpam-4396	191	1	(	(	PUNCT
ejpam-4396	191	2	a.	a.	NOUN
ejpam-4396	191	3	)	)	PUNCT
ejpam-4396	191	4	since	since	SCONJ
ejpam-4396	191	5	f	f	PROPN
ejpam-4396	191	6	is	be	AUX
ejpam-4396	191	7	a	a	DET
ejpam-4396	191	8	homomorphism	homomorphism	NOUN
ejpam-4396	191	9	,	,	PUNCT
ejpam-4396	191	10	we	we	PRON
ejpam-4396	191	11	have	have	VERB
ejpam-4396	191	12	f(a)f(e	f(a)f(e	ADV
ejpam-4396	191	13	)	)	PUNCT
ejpam-4396	191	14	=	=	SYM
ejpam-4396	191	15	f(ae	f(ae	NOUN
ejpam-4396	191	16	)	)	PUNCT
ejpam-4396	191	17	=	=	SYM
ejpam-4396	191	18	f(a	f(a	PROPN
ejpam-4396	191	19	)	)	PUNCT
ejpam-4396	191	20	.	.	PUNCT
ejpam-4396	192	1	hence	hence	ADV
ejpam-4396	192	2	,	,	PUNCT
ejpam-4396	192	3	f(e	f(e	NOUN
ejpam-4396	192	4	)	)	PUNCT
ejpam-4396	192	5	is	be	AUX
ejpam-4396	192	6	also	also	ADV
ejpam-4396	192	7	an	an	DET
ejpam-4396	192	8	identity	identity	NOUN
ejpam-4396	192	9	of	of	ADP
ejpam-4396	192	10	f(a	f(a	NOUN
ejpam-4396	192	11	)	)	PUNCT
ejpam-4396	192	12	.	.	PUNCT
ejpam-4396	193	1	(	(	PUNCT
ejpam-4396	193	2	b.	b.	NOUN
ejpam-4396	193	3	)	)	PUNCT
ejpam-4396	193	4	in	in	ADP
ejpam-4396	193	5	the	the	DET
ejpam-4396	193	6	same	same	ADJ
ejpam-4396	193	7	token	token	NOUN
ejpam-4396	193	8	,	,	PUNCT
ejpam-4396	193	9	since	since	SCONJ
ejpam-4396	193	10	f	f	PROPN
ejpam-4396	193	11	is	be	AUX
ejpam-4396	193	12	a	a	DET
ejpam-4396	193	13	homomorphism	homomorphism	NOUN
ejpam-4396	193	14	,	,	PUNCT
ejpam-4396	193	15	we	we	PRON
ejpam-4396	193	16	have	have	AUX
ejpam-4396	193	17	f(e	f(e	NOUN
ejpam-4396	193	18	)	)	PUNCT
ejpam-4396	193	19	=	=	SYM
ejpam-4396	193	20	f(ab	f(ab	PROPN
ejpam-4396	193	21	)	)	PUNCT
ejpam-4396	193	22	=	=	SYM
ejpam-4396	193	23	f(a)f(b	f(a)f(b	NOUN
ejpam-4396	193	24	)	)	PUNCT
ejpam-4396	193	25	.	.	PUNCT
ejpam-4396	194	1	now	now	ADV
ejpam-4396	194	2	,	,	PUNCT
ejpam-4396	194	3	since	since	SCONJ
ejpam-4396	194	4	in	in	ADP
ejpam-4396	194	5	(	(	PUNCT
ejpam-4396	194	6	a.	a.	NOUN
ejpam-4396	194	7	)	)	PUNCT
ejpam-4396	194	8	f(e	f(e	NOUN
ejpam-4396	194	9	)	)	PUNCT
ejpam-4396	194	10	is	be	AUX
ejpam-4396	194	11	an	an	DET
ejpam-4396	194	12	identity	identity	NOUN
ejpam-4396	194	13	,	,	PUNCT
ejpam-4396	194	14	f(b	f(b	PROPN
ejpam-4396	194	15	)	)	PUNCT
ejpam-4396	194	16	is	be	AUX
ejpam-4396	194	17	also	also	ADV
ejpam-4396	194	18	an	an	DET
ejpam-4396	194	19	inverse	inverse	NOUN
ejpam-4396	194	20	of	of	ADP
ejpam-4396	194	21	f(a	f(a	PROPN
ejpam-4396	194	22	)	)	PUNCT
ejpam-4396	194	23	.	.	PUNCT
ejpam-4396	195	1	for	for	ADP
ejpam-4396	195	2	the	the	DET
ejpam-4396	195	3	next	next	ADJ
ejpam-4396	195	4	theorem	theorem	NOUN
ejpam-4396	195	5	,	,	PUNCT
ejpam-4396	195	6	theorem	theorem	VERB
ejpam-4396	195	7	5	5	NUM
ejpam-4396	195	8	,	,	PUNCT
ejpam-4396	195	9	we	we	PRON
ejpam-4396	195	10	let	let	VERB
ejpam-4396	195	11	g	g	PROPN
ejpam-4396	195	12	and	and	CCONJ
ejpam-4396	195	13	j	j	PROPN
ejpam-4396	195	14	be	be	VERB
ejpam-4396	195	15	abelian	abelian	ADJ
ejpam-4396	195	16	g	g	NOUN
ejpam-4396	195	17	-	-	PUNCT
ejpam-4396	195	18	groups	group	NOUN
ejpam-4396	195	19	.	.	PUNCT
ejpam-4396	196	1	in	in	ADP
ejpam-4396	196	2	addition	addition	NOUN
ejpam-4396	196	3	,	,	PUNCT
ejpam-4396	196	4	we	we	PRON
ejpam-4396	196	5	let	let	VERB
ejpam-4396	196	6	h	h	NOUN
ejpam-4396	196	7	=	=	PRON
ejpam-4396	196	8	{	{	PUNCT
ejpam-4396	196	9	h	h	NOUN
ejpam-4396	196	10	∈	∈	PROPN
ejpam-4396	196	11	g	g	PROPN
ejpam-4396	196	12	:	:	PUNCT
ejpam-4396	196	13	h	h	PROPN
ejpam-4396	196	14	is	be	AUX
ejpam-4396	196	15	a	a	DET
ejpam-4396	196	16	unit	unit	NOUN
ejpam-4396	196	17	}	}	PUNCT
ejpam-4396	196	18	and	and	CCONJ
ejpam-4396	196	19	h	h	NOUN
ejpam-4396	196	20	′	′	NUM
ejpam-4396	197	1	=	=	PUNCT
ejpam-4396	197	2	{	{	PUNCT
ejpam-4396	197	3	h	h	NOUN
ejpam-4396	197	4	∈	∈	PROPN
ejpam-4396	197	5	j	j	PROPN
ejpam-4396	197	6	:	:	PUNCT
ejpam-4396	198	1	h	h	PROPN
ejpam-4396	198	2	is	be	AUX
ejpam-4396	198	3	a	a	DET
ejpam-4396	198	4	unit	unit	NOUN
ejpam-4396	198	5	}	}	PUNCT
ejpam-4396	198	6	.	.	PUNCT
ejpam-4396	199	1	theorem	theorem	NOUN
ejpam-4396	199	2	5	5	NUM
ejpam-4396	199	3	.	.	PUNCT
ejpam-4396	200	1	if	if	SCONJ
ejpam-4396	200	2	both	both	DET
ejpam-4396	200	3	f	f	X
ejpam-4396	200	4	:	:	PUNCT
ejpam-4396	200	5	g	g	PROPN
ejpam-4396	200	6	→	→	SYM
ejpam-4396	200	7	j	j	PROPN
ejpam-4396	200	8	and	and	CCONJ
ejpam-4396	200	9	f−1	f−1	PROPN
ejpam-4396	200	10	:	:	PUNCT
ejpam-4396	200	11	j	j	PROPN
ejpam-4396	200	12	→	→	SYM
ejpam-4396	200	13	g	g	PROPN
ejpam-4396	200	14	are	be	AUX
ejpam-4396	200	15	homomorphisms	homomorphism	NOUN
ejpam-4396	200	16	,	,	PUNCT
ejpam-4396	200	17	then	then	ADV
ejpam-4396	200	18	h	h	PROPN
ejpam-4396	200	19	∈	∈	PROPN
ejpam-4396	200	20	h	h	NOUN
ejpam-4396	200	21	if	if	SCONJ
ejpam-4396	200	22	and	and	CCONJ
ejpam-4396	200	23	only	only	ADV
ejpam-4396	200	24	if	if	SCONJ
ejpam-4396	200	25	f(h	f(h	PROPN
ejpam-4396	200	26	)	)	PUNCT
ejpam-4396	200	27	∈	∈	PROPN
ejpam-4396	200	28	h	h	NOUN
ejpam-4396	200	29	′	′	NOUN
ejpam-4396	200	30	,	,	PUNCT
ejpam-4396	200	31	that	that	PRON
ejpam-4396	200	32	is	be	AUX
ejpam-4396	200	33	f(h	f(h	PROPN
ejpam-4396	200	34	)	)	PUNCT
ejpam-4396	201	1	=	=	SYM
ejpam-4396	201	2	h	h	NOUN
ejpam-4396	201	3	′.	′.	NOUN
ejpam-4396	201	4	proof	proof	NOUN
ejpam-4396	201	5	.	.	PUNCT
ejpam-4396	202	1	it	it	PRON
ejpam-4396	202	2	suffices	suffice	VERB
ejpam-4396	202	3	to	to	PART
ejpam-4396	202	4	show	show	VERB
ejpam-4396	202	5	that	that	SCONJ
ejpam-4396	202	6	if	if	SCONJ
ejpam-4396	202	7	h	h	PROPN
ejpam-4396	202	8	∈	∈	PROPN
ejpam-4396	202	9	h	h	NOUN
ejpam-4396	202	10	,	,	PUNCT
ejpam-4396	202	11	then	then	ADV
ejpam-4396	202	12	f(h	f(h	PROPN
ejpam-4396	202	13	)	)	PUNCT
ejpam-4396	202	14	∈	∈	PROPN
ejpam-4396	202	15	h	h	NOUN
ejpam-4396	202	16	′.	′.	NOUN
ejpam-4396	202	17	assume	assume	VERB
ejpam-4396	202	18	that	that	SCONJ
ejpam-4396	202	19	h	h	NOUN
ejpam-4396	202	20	∈	∈	PROPN
ejpam-4396	202	21	h	h	NOUN
ejpam-4396	202	22	and	and	CCONJ
ejpam-4396	202	23	f(h	f(h	PROPN
ejpam-4396	202	24	)	)	PUNCT
ejpam-4396	202	25	/∈	/∈	PUNCT
ejpam-4396	203	1	h	h	NOUN
ejpam-4396	203	2	′.	′.	NOUN
ejpam-4396	203	3	by	by	ADP
ejpam-4396	203	4	theorem	theorem	NOUN
ejpam-4396	203	5	4(a	4(a	NUM
ejpam-4396	203	6	)	)	PUNCT
ejpam-4396	203	7	,	,	PUNCT
ejpam-4396	203	8	f(eh	f(eh	NUM
ejpam-4396	203	9	)	)	PUNCT
ejpam-4396	203	10	is	be	AUX
ejpam-4396	203	11	an	an	DET
ejpam-4396	203	12	identity	identity	NOUN
ejpam-4396	203	13	of	of	ADP
ejpam-4396	203	14	f(h	f(h	PROPN
ejpam-4396	203	15	)	)	PUNCT
ejpam-4396	203	16	.	.	PUNCT
ejpam-4396	204	1	if	if	SCONJ
ejpam-4396	204	2	f(h	f(h	PROPN
ejpam-4396	204	3	)	)	PUNCT
ejpam-4396	204	4	/∈	/∈	PUNCT
ejpam-4396	205	1	h	h	NOUN
ejpam-4396	205	2	′	′	NOUN
ejpam-4396	205	3	,	,	PUNCT
ejpam-4396	205	4	then	then	ADV
ejpam-4396	205	5	there	there	PRON
ejpam-4396	205	6	is	be	VERB
ejpam-4396	205	7	another	another	DET
ejpam-4396	205	8	identity	identity	NOUN
ejpam-4396	205	9	of	of	ADP
ejpam-4396	205	10	f(h	f(h	PROPN
ejpam-4396	205	11	)	)	PUNCT
ejpam-4396	205	12	,	,	PUNCT
ejpam-4396	205	13	say	say	VERB
ejpam-4396	205	14	e	e	NOUN
ejpam-4396	205	15	,	,	PUNCT
ejpam-4396	205	16	in	in	ADP
ejpam-4396	205	17	j	j	PROPN
ejpam-4396	205	18	with	with	ADP
ejpam-4396	205	19	e	e	PROPN
ejpam-4396	205	20	̸=	̸=	PROPN
ejpam-4396	205	21	f(eh	f(eh	NUM
ejpam-4396	205	22	)	)	PUNCT
ejpam-4396	205	23	.	.	PUNCT
ejpam-4396	206	1	since	since	SCONJ
ejpam-4396	206	2	f−1	f−1	PROPN
ejpam-4396	206	3	is	be	AUX
ejpam-4396	206	4	also	also	ADV
ejpam-4396	206	5	a	a	DET
ejpam-4396	206	6	homomorphism	homomorphism	NOUN
ejpam-4396	206	7	,	,	PUNCT
ejpam-4396	206	8	we	we	PRON
ejpam-4396	206	9	have	have	VERB
ejpam-4396	206	10	h	h	NOUN
ejpam-4396	206	11	=	=	SYM
ejpam-4396	206	12	f−1(f(h	f−1(f(h	ADJ
ejpam-4396	206	13	)	)	PUNCT
ejpam-4396	206	14	)	)	PUNCT
ejpam-4396	207	1	=	=	SYM
ejpam-4396	207	2	f−1(f(h)e	f−1(f(h)e	NOUN
ejpam-4396	207	3	)	)	PUNCT
ejpam-4396	207	4	=	=	SYM
ejpam-4396	207	5	f−1(f(h))f−1(e	f−1(f(h))f−1(e	X
ejpam-4396	207	6	)	)	PUNCT
ejpam-4396	207	7	=	=	SYM
ejpam-4396	207	8	hf−1(e	hf−1(e	NOUN
ejpam-4396	207	9	)	)	PUNCT
ejpam-4396	207	10	,	,	PUNCT
ejpam-4396	207	11	that	that	PRON
ejpam-4396	207	12	is	be	AUX
ejpam-4396	207	13	f−1(e	f−1(e	NOUN
ejpam-4396	207	14	)	)	PUNCT
ejpam-4396	207	15	is	be	AUX
ejpam-4396	207	16	another	another	DET
ejpam-4396	207	17	identity	identity	NOUN
ejpam-4396	207	18	of	of	ADP
ejpam-4396	207	19	h.	h.	NOUN
ejpam-4396	207	20	this	this	PRON
ejpam-4396	207	21	is	be	AUX
ejpam-4396	207	22	a	a	DET
ejpam-4396	207	23	contradiction	contradiction	NOUN
ejpam-4396	207	24	.	.	PUNCT
ejpam-4396	208	1	remark	remark	NOUN
ejpam-4396	208	2	8	8	NUM
ejpam-4396	208	3	maybe	maybe	ADV
ejpam-4396	208	4	worth	worth	ADJ
ejpam-4396	208	5	noting	note	VERB
ejpam-4396	208	6	.	.	PUNCT
ejpam-4396	209	1	we	we	PRON
ejpam-4396	209	2	say	say	VERB
ejpam-4396	209	3	that	that	SCONJ
ejpam-4396	209	4	a	a	DET
ejpam-4396	209	5	homomorphism	homomorphism	NOUN
ejpam-4396	209	6	is	be	AUX
ejpam-4396	209	7	a	a	DET
ejpam-4396	209	8	monomorphism	monomorphism	NOUN
ejpam-4396	209	9	if	if	SCONJ
ejpam-4396	209	10	it	it	PRON
ejpam-4396	209	11	is	be	AUX
ejpam-4396	209	12	injective	injective	ADJ
ejpam-4396	209	13	.	.	PUNCT
ejpam-4396	210	1	remark	remark	PROPN
ejpam-4396	210	2	8	8	NUM
ejpam-4396	210	3	.	.	PUNCT
ejpam-4396	211	1	let	let	VERB
ejpam-4396	211	2	g	g	PROPN
ejpam-4396	211	3	and	and	CCONJ
ejpam-4396	211	4	j	j	PROPN
ejpam-4396	211	5	be	be	VERB
ejpam-4396	211	6	abelian	abelian	ADJ
ejpam-4396	211	7	g	g	NOUN
ejpam-4396	211	8	-	-	PUNCT
ejpam-4396	211	9	groups	group	NOUN
ejpam-4396	211	10	,	,	PUNCT
ejpam-4396	211	11	and	and	CCONJ
ejpam-4396	211	12	f	f	X
ejpam-4396	211	13	:	:	PUNCT
ejpam-4396	211	14	g	g	PROPN
ejpam-4396	211	15	→	→	SYM
ejpam-4396	211	16	j	j	PROPN
ejpam-4396	211	17	be	be	AUX
ejpam-4396	211	18	a	a	DET
ejpam-4396	211	19	monomorphism	monomorphism	NOUN
ejpam-4396	211	20	.	.	PUNCT
ejpam-4396	212	1	if	if	SCONJ
ejpam-4396	212	2	g	g	PROPN
ejpam-4396	212	3	is	be	AUX
ejpam-4396	212	4	not	not	PART
ejpam-4396	212	5	unit	unit	NOUN
ejpam-4396	212	6	of	of	ADP
ejpam-4396	212	7	g	g	NOUN
ejpam-4396	212	8	,	,	PUNCT
ejpam-4396	212	9	then	then	ADV
ejpam-4396	212	10	f(g	f(g	NOUN
ejpam-4396	212	11	)	)	PUNCT
ejpam-4396	212	12	is	be	AUX
ejpam-4396	212	13	not	not	PART
ejpam-4396	212	14	a	a	DET
ejpam-4396	212	15	unit	unit	NOUN
ejpam-4396	212	16	of	of	ADP
ejpam-4396	212	17	j	j	PROPN
ejpam-4396	212	18	.	.	PUNCT
ejpam-4396	213	1	to	to	PART
ejpam-4396	213	2	see	see	VERB
ejpam-4396	213	3	this	this	PRON
ejpam-4396	213	4	,	,	PUNCT
ejpam-4396	213	5	let	let	VERB
ejpam-4396	213	6	e	e	NOUN
ejpam-4396	213	7	and	and	CCONJ
ejpam-4396	213	8	e′	e′	VERB
ejpam-4396	213	9	be	be	AUX
ejpam-4396	213	10	distinct	distinct	ADJ
ejpam-4396	213	11	identities	identity	NOUN
ejpam-4396	213	12	of	of	ADP
ejpam-4396	213	13	a.	a.	NOUN
ejpam-4396	213	14	by	by	ADP
ejpam-4396	213	15	theorem	theorem	NOUN
ejpam-4396	213	16	4(a	4(a	NUM
ejpam-4396	213	17	)	)	PUNCT
ejpam-4396	213	18	,	,	PUNCT
ejpam-4396	213	19	f(e	f(e	NOUN
ejpam-4396	213	20	)	)	PUNCT
ejpam-4396	213	21	and	and	CCONJ
ejpam-4396	213	22	f(e′	f(e′	NOUN
ejpam-4396	213	23	)	)	PUNCT
ejpam-4396	213	24	are	be	AUX
ejpam-4396	213	25	identities	identity	NOUN
ejpam-4396	213	26	of	of	ADP
ejpam-4396	213	27	f(a	f(a	NOUN
ejpam-4396	213	28	)	)	PUNCT
ejpam-4396	213	29	.	.	PUNCT
ejpam-4396	214	1	since	since	SCONJ
ejpam-4396	214	2	f	f	PROPN
ejpam-4396	214	3	is	be	AUX
ejpam-4396	214	4	injective	injective	ADJ
ejpam-4396	214	5	,	,	PUNCT
ejpam-4396	214	6	f(e	f(e	NOUN
ejpam-4396	214	7	)	)	PUNCT
ejpam-4396	214	8	̸=	̸=	PROPN
ejpam-4396	214	9	f(e′	f(e′	NOUN
ejpam-4396	214	10	)	)	PUNCT
ejpam-4396	214	11	.	.	PUNCT
ejpam-4396	215	1	thus	thus	ADV
ejpam-4396	215	2	,	,	PUNCT
ejpam-4396	215	3	f(a	f(a	PROPN
ejpam-4396	215	4	)	)	PUNCT
ejpam-4396	215	5	is	be	AUX
ejpam-4396	215	6	not	not	PART
ejpam-4396	215	7	a	a	DET
ejpam-4396	215	8	unit	unit	NOUN
ejpam-4396	215	9	.	.	PUNCT
ejpam-4396	216	1	the	the	DET
ejpam-4396	216	2	next	next	ADJ
ejpam-4396	216	3	statement	statement	NOUN
ejpam-4396	216	4	,	,	PUNCT
ejpam-4396	216	5	corollary	corollary	ADJ
ejpam-4396	216	6	3	3	NUM
ejpam-4396	216	7	,	,	PUNCT
ejpam-4396	216	8	follows	follow	VERB
ejpam-4396	216	9	from	from	ADP
ejpam-4396	216	10	remark	remark	NOUN
ejpam-4396	216	11	8	8	NUM
ejpam-4396	216	12	.	.	PUNCT
ejpam-4396	217	1	we	we	PRON
ejpam-4396	217	2	let	let	VERB
ejpam-4396	217	3	h	h	NOUN
ejpam-4396	217	4	=	=	PRON
ejpam-4396	217	5	{	{	PUNCT
ejpam-4396	217	6	h	h	NOUN
ejpam-4396	217	7	∈	∈	PROPN
ejpam-4396	217	8	g	g	PROPN
ejpam-4396	217	9	:	:	PUNCT
ejpam-4396	217	10	h	h	PROPN
ejpam-4396	217	11	is	be	AUX
ejpam-4396	217	12	a	a	DET
ejpam-4396	217	13	unit	unit	NOUN
ejpam-4396	217	14	}	}	PUNCT
ejpam-4396	217	15	.	.	PUNCT
ejpam-4396	218	1	j.	j.	PROPN
ejpam-4396	218	2	caraquil	caraquil	PROPN
ejpam-4396	218	3	,	,	PUNCT
ejpam-4396	218	4	m.	m.	PROPN
ejpam-4396	218	5	baldado	baldado	PROPN
ejpam-4396	218	6	jr	jr	PROPN
ejpam-4396	218	7	.	.	PROPN
ejpam-4396	218	8	/	/	SYM
ejpam-4396	218	9	eur	eur	PROPN
ejpam-4396	218	10	.	.	PUNCT
ejpam-4396	219	1	j.	j.	PROPN
ejpam-4396	219	2	pure	pure	PROPN
ejpam-4396	219	3	appl	appl	PROPN
ejpam-4396	219	4	.	.	PROPN
ejpam-4396	219	5	math	math	PROPN
ejpam-4396	219	6	,	,	PUNCT
ejpam-4396	219	7	15	15	NUM
ejpam-4396	219	8	(	(	PUNCT
ejpam-4396	219	9	3	3	NUM
ejpam-4396	219	10	)	)	PUNCT
ejpam-4396	219	11	(	(	PUNCT
ejpam-4396	219	12	2022	2022	NUM
ejpam-4396	219	13	)	)	PUNCT
ejpam-4396	219	14	,	,	PUNCT
ejpam-4396	219	15	887	887	NUM
ejpam-4396	219	16	-	-	SYM
ejpam-4396	219	17	896	896	NUM
ejpam-4396	219	18	894	894	NUM
ejpam-4396	219	19	corollary	corollary	NOUN
ejpam-4396	219	20	3	3	NUM
ejpam-4396	219	21	.	.	PUNCT
ejpam-4396	220	1	let	let	VERB
ejpam-4396	220	2	g	g	NOUN
ejpam-4396	220	3	be	be	AUX
ejpam-4396	220	4	an	an	DET
ejpam-4396	220	5	abelian	abelian	ADJ
ejpam-4396	220	6	g	g	NOUN
ejpam-4396	220	7	-	-	PUNCT
ejpam-4396	220	8	group	group	NOUN
ejpam-4396	220	9	,	,	PUNCT
ejpam-4396	220	10	and	and	CCONJ
ejpam-4396	220	11	f	f	X
ejpam-4396	220	12	:	:	PUNCT
ejpam-4396	220	13	g	g	PROPN
ejpam-4396	220	14	→	→	SYM
ejpam-4396	220	15	g	g	NOUN
ejpam-4396	220	16	be	be	AUX
ejpam-4396	220	17	a	a	DET
ejpam-4396	220	18	monomorphism	monomorphism	NOUN
ejpam-4396	220	19	.	.	PUNCT
ejpam-4396	221	1	if	if	SCONJ
ejpam-4396	221	2	a	a	DET
ejpam-4396	221	3	∈	∈	NOUN
ejpam-4396	221	4	g\h	g\h	NOUN
ejpam-4396	221	5	,	,	PUNCT
ejpam-4396	221	6	then	then	ADV
ejpam-4396	221	7	so	so	ADV
ejpam-4396	221	8	is	be	AUX
ejpam-4396	221	9	f(a	f(a	PROPN
ejpam-4396	221	10	)	)	PUNCT
ejpam-4396	221	11	.	.	PUNCT
ejpam-4396	222	1	the	the	DET
ejpam-4396	222	2	next	next	ADJ
ejpam-4396	222	3	statement	statement	NOUN
ejpam-4396	222	4	,	,	PUNCT
ejpam-4396	222	5	theorem	theorem	VERB
ejpam-4396	222	6	6	6	NUM
ejpam-4396	222	7	,	,	PUNCT
ejpam-4396	222	8	also	also	ADV
ejpam-4396	222	9	provides	provide	VERB
ejpam-4396	222	10	a	a	DET
ejpam-4396	222	11	way	way	NOUN
ejpam-4396	222	12	of	of	ADP
ejpam-4396	222	13	constructing	construct	VERB
ejpam-4396	222	14	a	a	DET
ejpam-4396	222	15	g	g	NOUN
ejpam-4396	222	16	-	-	PUNCT
ejpam-4396	222	17	subgroups	subgroup	NOUN
ejpam-4396	222	18	via	via	ADP
ejpam-4396	222	19	a	a	DET
ejpam-4396	222	20	homomorphism	homomorphism	NOUN
ejpam-4396	222	21	.	.	PUNCT
ejpam-4396	223	1	theorem	theorem	ADJ
ejpam-4396	223	2	6	6	NUM
ejpam-4396	223	3	.	.	PUNCT
ejpam-4396	224	1	let	let	VERB
ejpam-4396	224	2	g	g	PROPN
ejpam-4396	224	3	and	and	CCONJ
ejpam-4396	224	4	j	j	PROPN
ejpam-4396	224	5	be	be	VERB
ejpam-4396	224	6	abelian	abelian	ADJ
ejpam-4396	224	7	g	g	NOUN
ejpam-4396	224	8	-	-	PUNCT
ejpam-4396	224	9	groups	group	NOUN
ejpam-4396	224	10	.	.	PUNCT
ejpam-4396	225	1	if	if	SCONJ
ejpam-4396	225	2	f	f	PROPN
ejpam-4396	225	3	:	:	PUNCT
ejpam-4396	225	4	g	g	PROPN
ejpam-4396	225	5	→	→	SYM
ejpam-4396	225	6	j	j	PROPN
ejpam-4396	225	7	is	be	AUX
ejpam-4396	225	8	a	a	DET
ejpam-4396	225	9	homomorphism	homomorphism	NOUN
ejpam-4396	225	10	,	,	PUNCT
ejpam-4396	225	11	then	then	ADV
ejpam-4396	225	12	f(g	f(g	NOUN
ejpam-4396	225	13	)	)	PUNCT
ejpam-4396	225	14	abelian	abelian	NOUN
ejpam-4396	225	15	g	g	PROPN
ejpam-4396	225	16	-	-	PUNCT
ejpam-4396	225	17	subgroup	subgroup	NOUN
ejpam-4396	225	18	of	of	ADP
ejpam-4396	225	19	j	j	PROPN
ejpam-4396	225	20	.	.	PUNCT
ejpam-4396	226	1	proof	proof	NOUN
ejpam-4396	226	2	.	.	PUNCT
ejpam-4396	227	1	since	since	SCONJ
ejpam-4396	227	2	the	the	DET
ejpam-4396	227	3	operation	operation	NOUN
ejpam-4396	227	4	is	be	AUX
ejpam-4396	227	5	closed	closed	ADJ
ejpam-4396	227	6	,	,	PUNCT
ejpam-4396	227	7	associative	associative	ADJ
ejpam-4396	227	8	,	,	PUNCT
ejpam-4396	227	9	and	and	CCONJ
ejpam-4396	227	10	commutative	commutative	ADJ
ejpam-4396	227	11	in	in	ADP
ejpam-4396	227	12	g	g	PROPN
ejpam-4396	227	13	,	,	PUNCT
ejpam-4396	227	14	the	the	DET
ejpam-4396	227	15	conditions	condition	NOUN
ejpam-4396	227	16	for	for	ADP
ejpam-4396	227	17	homomorphism	homomorphism	NOUN
ejpam-4396	227	18	should	should	AUX
ejpam-4396	227	19	imply	imply	VERB
ejpam-4396	227	20	the	the	DET
ejpam-4396	227	21	closure	closure	NOUN
ejpam-4396	227	22	,	,	PUNCT
ejpam-4396	227	23	the	the	DET
ejpam-4396	227	24	associativity	associativity	NOUN
ejpam-4396	227	25	,	,	PUNCT
ejpam-4396	227	26	and	and	CCONJ
ejpam-4396	227	27	the	the	DET
ejpam-4396	227	28	commutativity	commutativity	NOUN
ejpam-4396	227	29	of	of	ADP
ejpam-4396	227	30	the	the	DET
ejpam-4396	227	31	operation	operation	NOUN
ejpam-4396	227	32	in	in	ADP
ejpam-4396	227	33	f(g	f(g	NOUN
ejpam-4396	227	34	)	)	PUNCT
ejpam-4396	227	35	.	.	PUNCT
ejpam-4396	228	1	hence	hence	ADV
ejpam-4396	228	2	,	,	PUNCT
ejpam-4396	228	3	the	the	DET
ejpam-4396	228	4	closure	closure	NOUN
ejpam-4396	228	5	property	property	NOUN
ejpam-4396	228	6	,	,	PUNCT
ejpam-4396	228	7	stated	state	VERB
ejpam-4396	228	8	in	in	ADP
ejpam-4396	228	9	(	(	PUNCT
ejpam-4396	228	10	g1	g1	PROPN
ejpam-4396	228	11	)	)	PUNCT
ejpam-4396	228	12	,	,	PUNCT
ejpam-4396	228	13	and	and	CCONJ
ejpam-4396	228	14	the	the	DET
ejpam-4396	228	15	commutativity	commutativity	NOUN
ejpam-4396	228	16	requirements	requirement	NOUN
ejpam-4396	228	17	are	be	AUX
ejpam-4396	228	18	satisfied	satisfied	ADJ
ejpam-4396	228	19	.	.	PUNCT
ejpam-4396	229	1	finally	finally	ADV
ejpam-4396	229	2	,	,	PUNCT
ejpam-4396	229	3	(	(	PUNCT
ejpam-4396	229	4	g2	g2	PROPN
ejpam-4396	229	5	)	)	PUNCT
ejpam-4396	229	6	and	and	CCONJ
ejpam-4396	229	7	(	(	PUNCT
ejpam-4396	229	8	g3	g3	NOUN
ejpam-4396	229	9	)	)	PUNCT
ejpam-4396	229	10	follow	follow	VERB
ejpam-4396	229	11	from	from	ADP
ejpam-4396	229	12	theorem	theorem	ADJ
ejpam-4396	229	13	4	4	NUM
ejpam-4396	229	14	.	.	NOUN
ejpam-4396	229	15	5	5	NUM
ejpam-4396	229	16	.	.	PUNCT
ejpam-4396	230	1	the	the	DET
ejpam-4396	230	2	zero	zero	NUM
ejpam-4396	230	3	element	element	NOUN
ejpam-4396	230	4	in	in	ADP
ejpam-4396	230	5	this	this	DET
ejpam-4396	230	6	section	section	NOUN
ejpam-4396	230	7	,	,	PUNCT
ejpam-4396	230	8	we	we	PRON
ejpam-4396	230	9	give	give	VERB
ejpam-4396	230	10	some	some	DET
ejpam-4396	230	11	important	important	ADJ
ejpam-4396	230	12	properties	property	NOUN
ejpam-4396	230	13	of	of	ADP
ejpam-4396	230	14	zero	zero	NUM
ejpam-4396	230	15	elements	element	NOUN
ejpam-4396	230	16	and	and	CCONJ
ejpam-4396	230	17	zero	zero	NUM
ejpam-4396	230	18	-	-	PUNCT
ejpam-4396	230	19	divisors	divisor	NOUN
ejpam-4396	230	20	.	.	PUNCT
ejpam-4396	231	1	at	at	ADP
ejpam-4396	231	2	the	the	DET
ejpam-4396	231	3	end	end	NOUN
ejpam-4396	231	4	of	of	ADP
ejpam-4396	231	5	this	this	DET
ejpam-4396	231	6	section	section	NOUN
ejpam-4396	231	7	,	,	PUNCT
ejpam-4396	231	8	we	we	PRON
ejpam-4396	231	9	presented	present	VERB
ejpam-4396	231	10	two	two	NUM
ejpam-4396	231	11	corollaries	corollary	NOUN
ejpam-4396	231	12	that	that	PRON
ejpam-4396	231	13	provide	provide	VERB
ejpam-4396	231	14	another	another	DET
ejpam-4396	231	15	way	way	NOUN
ejpam-4396	231	16	of	of	ADP
ejpam-4396	231	17	constructing	construct	VERB
ejpam-4396	231	18	a	a	DET
ejpam-4396	231	19	g	g	NOUN
ejpam-4396	231	20	-	-	PUNCT
ejpam-4396	231	21	subgroup	subgroup	NOUN
ejpam-4396	231	22	.	.	PUNCT
ejpam-4396	232	1	the	the	DET
ejpam-4396	232	2	next	next	ADJ
ejpam-4396	232	3	statement	statement	NOUN
ejpam-4396	232	4	,	,	PUNCT
ejpam-4396	232	5	definition	definition	NOUN
ejpam-4396	232	6	3	3	NUM
ejpam-4396	232	7	,	,	PUNCT
ejpam-4396	232	8	describes	describe	VERB
ejpam-4396	232	9	what	what	PRON
ejpam-4396	232	10	a	a	DET
ejpam-4396	232	11	zero	zero	NUM
ejpam-4396	232	12	element	element	NOUN
ejpam-4396	232	13	is	be	AUX
ejpam-4396	232	14	.	.	PUNCT
ejpam-4396	233	1	definition	definition	NOUN
ejpam-4396	233	2	3	3	NUM
ejpam-4396	233	3	.	.	PUNCT
ejpam-4396	234	1	let	let	VERB
ejpam-4396	234	2	g	g	PRON
ejpam-4396	234	3	be	be	AUX
ejpam-4396	234	4	a	a	DET
ejpam-4396	234	5	g	g	NOUN
ejpam-4396	234	6	-	-	PUNCT
ejpam-4396	234	7	group	group	NOUN
ejpam-4396	234	8	.	.	PUNCT
ejpam-4396	235	1	an	an	DET
ejpam-4396	235	2	element	element	NOUN
ejpam-4396	235	3	0	0	NUM
ejpam-4396	235	4	∈	∈	PROPN
ejpam-4396	235	5	g	g	PROPN
ejpam-4396	235	6	is	be	AUX
ejpam-4396	235	7	called	call	VERB
ejpam-4396	235	8	a	a	DET
ejpam-4396	235	9	zero	zero	NUM
ejpam-4396	235	10	if	if	SCONJ
ejpam-4396	235	11	a0	a0	PROPN
ejpam-4396	235	12	=	=	SYM
ejpam-4396	235	13	0a	0a	PROPN
ejpam-4396	236	1	=	=	SYM
ejpam-4396	236	2	0	0	NUM
ejpam-4396	236	3	for	for	ADP
ejpam-4396	236	4	all	all	DET
ejpam-4396	236	5	a	a	DET
ejpam-4396	236	6	∈	∈	PROPN
ejpam-4396	236	7	g.	g.	NOUN
ejpam-4396	236	8	example	example	NOUN
ejpam-4396	236	9	3	3	X
ejpam-4396	236	10	.	.	X
ejpam-4396	236	11	consider	consider	VERB
ejpam-4396	236	12	the	the	DET
ejpam-4396	236	13	g	g	NOUN
ejpam-4396	236	14	-	-	PUNCT
ejpam-4396	236	15	group	group	NOUN
ejpam-4396	236	16	(	(	PUNCT
ejpam-4396	236	17	s6,×6	s6,×6	PROPN
ejpam-4396	236	18	)	)	PUNCT
ejpam-4396	236	19	in	in	ADP
ejpam-4396	236	20	example	example	NOUN
ejpam-4396	237	1	1	1	X
ejpam-4396	237	2	.	.	X
ejpam-4396	237	3	note	note	VERB
ejpam-4396	237	4	that	that	SCONJ
ejpam-4396	237	5	the	the	DET
ejpam-4396	237	6	element	element	NOUN
ejpam-4396	237	7	0	0	NUM
ejpam-4396	237	8	is	be	AUX
ejpam-4396	237	9	the	the	DET
ejpam-4396	237	10	zero	zero	NUM
ejpam-4396	237	11	.	.	PUNCT
ejpam-4396	237	12	remark	remark	NOUN
ejpam-4396	237	13	9	9	NUM
ejpam-4396	237	14	.	.	PUNCT
ejpam-4396	238	1	a	a	DET
ejpam-4396	238	2	g	g	NOUN
ejpam-4396	238	3	-	-	PUNCT
ejpam-4396	238	4	group	group	NOUN
ejpam-4396	238	5	may	may	AUX
ejpam-4396	238	6	or	or	CCONJ
ejpam-4396	238	7	may	may	AUX
ejpam-4396	238	8	not	not	PART
ejpam-4396	238	9	have	have	VERB
ejpam-4396	238	10	a	a	DET
ejpam-4396	238	11	zero	zero	NUM
ejpam-4396	238	12	.	.	PUNCT
ejpam-4396	239	1	to	to	PART
ejpam-4396	239	2	see	see	VERB
ejpam-4396	239	3	this	this	PRON
ejpam-4396	239	4	,	,	PUNCT
ejpam-4396	239	5	consider	consider	VERB
ejpam-4396	239	6	again	again	ADV
ejpam-4396	239	7	the	the	DET
ejpam-4396	239	8	g	g	NOUN
ejpam-4396	239	9	-	-	PUNCT
ejpam-4396	239	10	group	group	NOUN
ejpam-4396	239	11	(	(	PUNCT
ejpam-4396	239	12	s6,×6	s6,×6	PROPN
ejpam-4396	239	13	)	)	PUNCT
ejpam-4396	239	14	in	in	ADP
ejpam-4396	239	15	example	example	NOUN
ejpam-4396	240	1	1	1	X
ejpam-4396	240	2	.	.	X
ejpam-4396	240	3	note	note	VERB
ejpam-4396	240	4	that	that	SCONJ
ejpam-4396	240	5	its	its	PRON
ejpam-4396	240	6	g	g	PROPN
ejpam-4396	240	7	-	-	PUNCT
ejpam-4396	240	8	subgroup	subgroup	NOUN
ejpam-4396	240	9	h	h	NOUN
ejpam-4396	240	10	=	=	PUNCT
ejpam-4396	240	11	{	{	PUNCT
ejpam-4396	240	12	1	1	NUM
ejpam-4396	240	13	,	,	PUNCT
ejpam-4396	240	14	5	5	NUM
ejpam-4396	240	15	}	}	PUNCT
ejpam-4396	240	16	does	do	AUX
ejpam-4396	240	17	not	not	PART
ejpam-4396	240	18	have	have	VERB
ejpam-4396	240	19	a	a	DET
ejpam-4396	240	20	zero	zero	NUM
ejpam-4396	240	21	element	element	NOUN
ejpam-4396	240	22	while	while	SCONJ
ejpam-4396	240	23	g\h	g\h	PROPN
ejpam-4396	240	24	has	have	VERB
ejpam-4396	240	25	.	.	PUNCT
ejpam-4396	241	1	it	it	PRON
ejpam-4396	241	2	is	be	AUX
ejpam-4396	241	3	easy	easy	ADJ
ejpam-4396	241	4	to	to	PART
ejpam-4396	241	5	see	see	VERB
ejpam-4396	241	6	that	that	SCONJ
ejpam-4396	241	7	a	a	DET
ejpam-4396	241	8	zero	zero	NUM
ejpam-4396	241	9	in	in	ADP
ejpam-4396	241	10	a	a	DET
ejpam-4396	241	11	non	non	ADJ
ejpam-4396	241	12	-	-	ADJ
ejpam-4396	241	13	trivial	trivial	ADJ
ejpam-4396	241	14	g	g	NOUN
ejpam-4396	241	15	-	-	PUNCT
ejpam-4396	241	16	group	group	NOUN
ejpam-4396	241	17	is	be	AUX
ejpam-4396	241	18	not	not	PART
ejpam-4396	241	19	a	a	DET
ejpam-4396	241	20	unit	unit	NOUN
ejpam-4396	241	21	,	,	PUNCT
ejpam-4396	241	22	since	since	SCONJ
ejpam-4396	241	23	all	all	DET
ejpam-4396	241	24	the	the	DET
ejpam-4396	241	25	other	other	ADJ
ejpam-4396	241	26	elements	element	NOUN
ejpam-4396	241	27	is	be	AUX
ejpam-4396	241	28	its	its	PRON
ejpam-4396	241	29	identity	identity	NOUN
ejpam-4396	241	30	.	.	PUNCT
ejpam-4396	242	1	the	the	DET
ejpam-4396	242	2	next	next	ADJ
ejpam-4396	242	3	statement	statement	NOUN
ejpam-4396	242	4	,	,	PUNCT
ejpam-4396	242	5	theorem	theorem	VERB
ejpam-4396	242	6	7	7	NUM
ejpam-4396	242	7	,	,	PUNCT
ejpam-4396	242	8	says	say	VERB
ejpam-4396	242	9	that	that	SCONJ
ejpam-4396	242	10	the	the	DET
ejpam-4396	242	11	zero	zero	NUM
ejpam-4396	242	12	element	element	NOUN
ejpam-4396	242	13	is	be	AUX
ejpam-4396	242	14	unique	unique	ADJ
ejpam-4396	242	15	if	if	SCONJ
ejpam-4396	242	16	it	it	PRON
ejpam-4396	242	17	exists	exist	VERB
ejpam-4396	242	18	.	.	PUNCT
ejpam-4396	243	1	theorem	theorem	VERB
ejpam-4396	243	2	7	7	NUM
ejpam-4396	243	3	.	.	PUNCT
ejpam-4396	244	1	a	a	DET
ejpam-4396	244	2	g	g	NOUN
ejpam-4396	244	3	-	-	PUNCT
ejpam-4396	244	4	group	group	NOUN
ejpam-4396	244	5	can	can	AUX
ejpam-4396	244	6	have	have	VERB
ejpam-4396	244	7	at	at	ADP
ejpam-4396	244	8	most	most	ADJ
ejpam-4396	244	9	one	one	NUM
ejpam-4396	244	10	zero	zero	NUM
ejpam-4396	244	11	element	element	NOUN
ejpam-4396	244	12	.	.	PUNCT
ejpam-4396	245	1	proof	proof	NOUN
ejpam-4396	245	2	.	.	PUNCT
ejpam-4396	246	1	let	let	VERB
ejpam-4396	246	2	g	g	PRON
ejpam-4396	246	3	be	be	AUX
ejpam-4396	246	4	a	a	DET
ejpam-4396	246	5	g	g	NOUN
ejpam-4396	246	6	-	-	PUNCT
ejpam-4396	246	7	group	group	NOUN
ejpam-4396	246	8	with	with	ADP
ejpam-4396	246	9	zeros	zero	NOUN
ejpam-4396	246	10	0	0	PUNCT
ejpam-4396	246	11	and	and	CCONJ
ejpam-4396	246	12	0′.	0′.	NOUN
ejpam-4396	246	13	then	then	ADV
ejpam-4396	246	14	a0	a0	PROPN
ejpam-4396	246	15	=	=	SYM
ejpam-4396	246	16	0	0	PUNCT
ejpam-4396	247	1	=	=	SYM
ejpam-4396	247	2	0a	0a	PROPN
ejpam-4396	247	3	and	and	CCONJ
ejpam-4396	247	4	a0′	a0′	NOUN
ejpam-4396	247	5	=	=	SYM
ejpam-4396	247	6	0′	0′	PUNCT
ejpam-4396	248	1	=	=	SYM
ejpam-4396	248	2	0′a	0′a	NOUN
ejpam-4396	248	3	for	for	ADP
ejpam-4396	248	4	all	all	DET
ejpam-4396	248	5	a	a	DET
ejpam-4396	248	6	∈	∈	PROPN
ejpam-4396	248	7	g.	g.	NOUN
ejpam-4396	248	8	thus	thus	ADV
ejpam-4396	248	9	,	,	PUNCT
ejpam-4396	248	10	0	0	X
ejpam-4396	248	11	=	=	SYM
ejpam-4396	248	12	00′	00′	PROPN
ejpam-4396	248	13	=	=	SYM
ejpam-4396	248	14	0′0	0′0	NOUN
ejpam-4396	249	1	=	=	NOUN
ejpam-4396	249	2	0	0	PROPN
ejpam-4396	249	3	.	.	PUNCT
ejpam-4396	250	1	the	the	DET
ejpam-4396	250	2	next	next	ADJ
ejpam-4396	250	3	statement	statement	NOUN
ejpam-4396	250	4	,	,	PUNCT
ejpam-4396	250	5	theorem	theorem	VERB
ejpam-4396	250	6	8	8	NUM
ejpam-4396	250	7	,	,	PUNCT
ejpam-4396	250	8	says	say	VERB
ejpam-4396	250	9	that	that	SCONJ
ejpam-4396	250	10	the	the	DET
ejpam-4396	250	11	image	image	NOUN
ejpam-4396	250	12	of	of	ADP
ejpam-4396	250	13	the	the	DET
ejpam-4396	250	14	zero	zero	NUM
ejpam-4396	250	15	element	element	NOUN
ejpam-4396	250	16	under	under	ADP
ejpam-4396	250	17	a	a	DET
ejpam-4396	250	18	homomorphism	homomorphism	NOUN
ejpam-4396	250	19	is	be	AUX
ejpam-4396	250	20	the	the	DET
ejpam-4396	250	21	zero	zero	NUM
ejpam-4396	250	22	element	element	NOUN
ejpam-4396	250	23	.	.	PUNCT
ejpam-4396	251	1	theorem	theorem	VERB
ejpam-4396	251	2	8	8	NUM
ejpam-4396	251	3	.	.	PUNCT
ejpam-4396	252	1	let	let	VERB
ejpam-4396	252	2	g	g	NOUN
ejpam-4396	252	3	and	and	CCONJ
ejpam-4396	252	4	g′	g′	NOUN
ejpam-4396	252	5	be	be	AUX
ejpam-4396	252	6	both	both	PRON
ejpam-4396	252	7	g	g	NOUN
ejpam-4396	252	8	-	-	PUNCT
ejpam-4396	252	9	groups	group	NOUN
ejpam-4396	252	10	with	with	ADP
ejpam-4396	252	11	zeros	zero	NOUN
ejpam-4396	252	12	0	0	NUM
ejpam-4396	252	13	and	and	CCONJ
ejpam-4396	252	14	0′	0′	NUM
ejpam-4396	252	15	,	,	PUNCT
ejpam-4396	252	16	respectively	respectively	ADV
ejpam-4396	252	17	.	.	PUNCT
ejpam-4396	253	1	if	if	SCONJ
ejpam-4396	253	2	f	f	PROPN
ejpam-4396	253	3	:	:	PUNCT
ejpam-4396	253	4	g	g	PROPN
ejpam-4396	253	5	→	→	SYM
ejpam-4396	253	6	g′	g′	NOUN
ejpam-4396	253	7	is	be	AUX
ejpam-4396	253	8	a	a	DET
ejpam-4396	253	9	homomorphism	homomorphism	NOUN
ejpam-4396	253	10	,	,	PUNCT
ejpam-4396	253	11	then	then	ADV
ejpam-4396	253	12	f(0	f(0	NOUN
ejpam-4396	253	13	)	)	PUNCT
ejpam-4396	253	14	=	=	NOUN
ejpam-4396	253	15	0′.	0′.	NOUN
ejpam-4396	253	16	proof	proof	NOUN
ejpam-4396	253	17	.	.	PUNCT
ejpam-4396	254	1	observe	observe	VERB
ejpam-4396	254	2	that	that	SCONJ
ejpam-4396	254	3	for	for	ADP
ejpam-4396	254	4	all	all	DET
ejpam-4396	254	5	a	a	DET
ejpam-4396	254	6	∈	∈	PROPN
ejpam-4396	254	7	g	g	NOUN
ejpam-4396	254	8	,	,	PUNCT
ejpam-4396	254	9	f(a)f(0	f(a)f(0	NOUN
ejpam-4396	254	10	)	)	PUNCT
ejpam-4396	254	11	=	=	PUNCT
ejpam-4396	254	12	f(a0	f(a0	X
ejpam-4396	254	13	)	)	PUNCT
ejpam-4396	254	14	=	=	SYM
ejpam-4396	254	15	f(0	f(0	NOUN
ejpam-4396	254	16	)	)	PUNCT
ejpam-4396	254	17	=	=	SYM
ejpam-4396	254	18	f(0a	f(0a	X
ejpam-4396	254	19	)	)	PUNCT
ejpam-4396	254	20	=	=	SYM
ejpam-4396	254	21	f(0)f(a	f(0)f(a	NOUN
ejpam-4396	254	22	)	)	PUNCT
ejpam-4396	254	23	.	.	PUNCT
ejpam-4396	255	1	the	the	DET
ejpam-4396	255	2	next	next	ADJ
ejpam-4396	255	3	statement	statement	NOUN
ejpam-4396	255	4	,	,	PUNCT
ejpam-4396	255	5	definition	definition	NOUN
ejpam-4396	255	6	4	4	NUM
ejpam-4396	255	7	,	,	PUNCT
ejpam-4396	255	8	describes	describe	VERB
ejpam-4396	255	9	what	what	PRON
ejpam-4396	255	10	a	a	DET
ejpam-4396	255	11	zero	zero	NUM
ejpam-4396	255	12	divisor	divisor	NOUN
ejpam-4396	255	13	is	be	AUX
ejpam-4396	255	14	.	.	PUNCT
ejpam-4396	256	1	j.	j.	PROPN
ejpam-4396	256	2	caraquil	caraquil	PROPN
ejpam-4396	256	3	,	,	PUNCT
ejpam-4396	256	4	m.	m.	PROPN
ejpam-4396	256	5	baldado	baldado	PROPN
ejpam-4396	256	6	jr	jr	PROPN
ejpam-4396	256	7	.	.	PROPN
ejpam-4396	256	8	/	/	SYM
ejpam-4396	256	9	eur	eur	PROPN
ejpam-4396	256	10	.	.	PUNCT
ejpam-4396	257	1	j.	j.	PROPN
ejpam-4396	257	2	pure	pure	PROPN
ejpam-4396	257	3	appl	appl	PROPN
ejpam-4396	257	4	.	.	PROPN
ejpam-4396	257	5	math	math	PROPN
ejpam-4396	257	6	,	,	PUNCT
ejpam-4396	257	7	15	15	NUM
ejpam-4396	257	8	(	(	PUNCT
ejpam-4396	257	9	3	3	NUM
ejpam-4396	257	10	)	)	PUNCT
ejpam-4396	257	11	(	(	PUNCT
ejpam-4396	257	12	2022	2022	NUM
ejpam-4396	257	13	)	)	PUNCT
ejpam-4396	257	14	,	,	PUNCT
ejpam-4396	257	15	887	887	NUM
ejpam-4396	257	16	-	-	SYM
ejpam-4396	257	17	896	896	NUM
ejpam-4396	257	18	895	895	NUM
ejpam-4396	257	19	definition	definition	NOUN
ejpam-4396	257	20	4	4	NUM
ejpam-4396	257	21	.	.	PUNCT
ejpam-4396	258	1	let	let	VERB
ejpam-4396	258	2	g	g	PRON
ejpam-4396	258	3	be	be	AUX
ejpam-4396	258	4	a	a	DET
ejpam-4396	258	5	g	g	NOUN
ejpam-4396	258	6	-	-	PUNCT
ejpam-4396	258	7	group	group	NOUN
ejpam-4396	258	8	with	with	ADP
ejpam-4396	258	9	a	a	DET
ejpam-4396	258	10	zero	zero	NUM
ejpam-4396	258	11	element	element	NOUN
ejpam-4396	258	12	0	0	NUM
ejpam-4396	258	13	,	,	PUNCT
ejpam-4396	258	14	and	and	CCONJ
ejpam-4396	258	15	a	a	DET
ejpam-4396	258	16	∈	∈	NOUN
ejpam-4396	258	17	g	g	NOUN
ejpam-4396	258	18	with	with	ADP
ejpam-4396	258	19	a	a	DET
ejpam-4396	258	20	̸=	̸=	PROPN
ejpam-4396	258	21	0	0	NUM
ejpam-4396	258	22	.	.	PUNCT
ejpam-4396	259	1	we	we	PRON
ejpam-4396	259	2	say	say	VERB
ejpam-4396	259	3	that	that	SCONJ
ejpam-4396	259	4	a	a	PRON
ejpam-4396	259	5	is	be	AUX
ejpam-4396	259	6	a	a	DET
ejpam-4396	259	7	zero	zero	NUM
ejpam-4396	259	8	divisor	divisor	NOUN
ejpam-4396	259	9	if	if	SCONJ
ejpam-4396	259	10	there	there	PRON
ejpam-4396	259	11	exists	exist	VERB
ejpam-4396	259	12	b	b	PROPN
ejpam-4396	259	13	∈	∈	PROPN
ejpam-4396	259	14	g	g	NOUN
ejpam-4396	259	15	with	with	ADP
ejpam-4396	259	16	b	b	PROPN
ejpam-4396	259	17	̸=	̸=	PROPN
ejpam-4396	259	18	0	0	NUM
ejpam-4396	259	19	such	such	ADJ
ejpam-4396	259	20	that	that	SCONJ
ejpam-4396	259	21	ab	ab	PROPN
ejpam-4396	259	22	=	=	SYM
ejpam-4396	259	23	0	0	PUNCT
ejpam-4396	259	24	=	=	SYM
ejpam-4396	259	25	ba	ba	PROPN
ejpam-4396	259	26	.	.	PUNCT
ejpam-4396	259	27	example	example	NOUN
ejpam-4396	259	28	4	4	NUM
ejpam-4396	259	29	.	.	PUNCT
ejpam-4396	260	1	in	in	ADP
ejpam-4396	260	2	the	the	DET
ejpam-4396	260	3	g	g	PROPN
ejpam-4396	260	4	-	-	PUNCT
ejpam-4396	260	5	group	group	NOUN
ejpam-4396	260	6	s6	s6	PROPN
ejpam-4396	260	7	=	=	SYM
ejpam-4396	260	8	{	{	PUNCT
ejpam-4396	260	9	0	0	NUM
ejpam-4396	260	10	,	,	PUNCT
ejpam-4396	260	11	1	1	NUM
ejpam-4396	260	12	,	,	PUNCT
ejpam-4396	260	13	2	2	NUM
ejpam-4396	260	14	,	,	PUNCT
ejpam-4396	260	15	3	3	NUM
ejpam-4396	260	16	,	,	PUNCT
ejpam-4396	260	17	4	4	NUM
ejpam-4396	260	18	,	,	PUNCT
ejpam-4396	260	19	5	5	NUM
ejpam-4396	260	20	}	}	PUNCT
ejpam-4396	260	21	,	,	PUNCT
ejpam-4396	260	22	2	2	NUM
ejpam-4396	260	23	,	,	PUNCT
ejpam-4396	260	24	3	3	NUM
ejpam-4396	260	25	and	and	CCONJ
ejpam-4396	260	26	4	4	NUM
ejpam-4396	260	27	are	be	AUX
ejpam-4396	260	28	zero	zero	NUM
ejpam-4396	260	29	divisors	divisor	NOUN
ejpam-4396	260	30	since	since	SCONJ
ejpam-4396	260	31	2(3	2(3	NUM
ejpam-4396	260	32	)	)	PUNCT
ejpam-4396	260	33	=	=	PUNCT
ejpam-4396	260	34	3(4	3(4	NUM
ejpam-4396	260	35	)	)	PUNCT
ejpam-4396	261	1	=	=	SYM
ejpam-4396	261	2	0	0	X
ejpam-4396	261	3	.	.	PUNCT
ejpam-4396	262	1	the	the	DET
ejpam-4396	262	2	next	next	ADJ
ejpam-4396	262	3	statement	statement	NOUN
ejpam-4396	262	4	,	,	PUNCT
ejpam-4396	262	5	theorem	theorem	VERB
ejpam-4396	262	6	9	9	NUM
ejpam-4396	262	7	,	,	PUNCT
ejpam-4396	262	8	says	say	VERB
ejpam-4396	262	9	that	that	SCONJ
ejpam-4396	262	10	the	the	DET
ejpam-4396	262	11	image	image	NOUN
ejpam-4396	262	12	of	of	ADP
ejpam-4396	262	13	a	a	DET
ejpam-4396	262	14	zero	zero	NUM
ejpam-4396	262	15	divisor	divisor	NOUN
ejpam-4396	262	16	under	under	ADP
ejpam-4396	262	17	a	a	DET
ejpam-4396	262	18	homomorphism	homomorphism	NOUN
ejpam-4396	262	19	is	be	AUX
ejpam-4396	262	20	a	a	DET
ejpam-4396	262	21	zero	zero	NUM
ejpam-4396	262	22	divisor	divisor	NOUN
ejpam-4396	262	23	.	.	PUNCT
ejpam-4396	263	1	theorem	theorem	VERB
ejpam-4396	263	2	9	9	NUM
ejpam-4396	263	3	.	.	PUNCT
ejpam-4396	264	1	let	let	VERB
ejpam-4396	264	2	g	g	NOUN
ejpam-4396	264	3	and	and	CCONJ
ejpam-4396	264	4	g′	g′	NOUN
ejpam-4396	264	5	be	be	AUX
ejpam-4396	264	6	both	both	PRON
ejpam-4396	264	7	g	g	NOUN
ejpam-4396	264	8	-	-	PUNCT
ejpam-4396	264	9	groups	group	NOUN
ejpam-4396	264	10	with	with	ADP
ejpam-4396	264	11	zeros	zero	NOUN
ejpam-4396	264	12	0	0	NUM
ejpam-4396	264	13	and	and	CCONJ
ejpam-4396	264	14	0′	0′	NUM
ejpam-4396	264	15	,	,	PUNCT
ejpam-4396	264	16	respectively	respectively	ADV
ejpam-4396	264	17	.	.	PUNCT
ejpam-4396	265	1	if	if	SCONJ
ejpam-4396	265	2	f	f	PROPN
ejpam-4396	265	3	:	:	PUNCT
ejpam-4396	265	4	g	g	PROPN
ejpam-4396	265	5	→	→	SYM
ejpam-4396	265	6	g′	g′	NOUN
ejpam-4396	265	7	is	be	AUX
ejpam-4396	265	8	a	a	DET
ejpam-4396	265	9	homomorphism	homomorphism	NOUN
ejpam-4396	265	10	,	,	PUNCT
ejpam-4396	265	11	then	then	ADV
ejpam-4396	265	12	the	the	DET
ejpam-4396	265	13	images	image	NOUN
ejpam-4396	265	14	of	of	ADP
ejpam-4396	265	15	the	the	DET
ejpam-4396	265	16	zero	zero	NUM
ejpam-4396	265	17	divisors	divisor	NOUN
ejpam-4396	265	18	in	in	ADP
ejpam-4396	265	19	g	g	PROPN
ejpam-4396	265	20	are	be	AUX
ejpam-4396	265	21	zero	zero	NUM
ejpam-4396	265	22	divisors	divisor	NOUN
ejpam-4396	265	23	in	in	ADP
ejpam-4396	265	24	g′.	g′.	ADP
ejpam-4396	265	25	proof	proof	NOUN
ejpam-4396	265	26	.	.	PUNCT
ejpam-4396	266	1	let	let	VERB
ejpam-4396	266	2	a	a	PRON
ejpam-4396	266	3	be	be	AUX
ejpam-4396	266	4	a	a	DET
ejpam-4396	266	5	zero	zero	NUM
ejpam-4396	266	6	divisor	divisor	NOUN
ejpam-4396	266	7	in	in	ADP
ejpam-4396	266	8	g.	g.	PROPN
ejpam-4396	266	9	then	then	ADV
ejpam-4396	266	10	there	there	PRON
ejpam-4396	266	11	exists	exist	VERB
ejpam-4396	266	12	b	b	PROPN
ejpam-4396	266	13	̸=	̸=	PROPN
ejpam-4396	266	14	0	0	NUM
ejpam-4396	266	15	such	such	ADJ
ejpam-4396	266	16	that	that	SCONJ
ejpam-4396	266	17	ab	ab	PROPN
ejpam-4396	266	18	=	=	SYM
ejpam-4396	266	19	0	0	PUNCT
ejpam-4396	266	20	=	=	SYM
ejpam-4396	266	21	ba	ba	PROPN
ejpam-4396	266	22	.	.	PUNCT
ejpam-4396	266	23	note	note	VERB
ejpam-4396	266	24	that	that	SCONJ
ejpam-4396	266	25	by	by	ADP
ejpam-4396	266	26	theorem	theorem	ADJ
ejpam-4396	266	27	8	8	NUM
ejpam-4396	266	28	,	,	PUNCT
ejpam-4396	266	29	f(0	f(0	NOUN
ejpam-4396	266	30	)	)	PUNCT
ejpam-4396	266	31	is	be	AUX
ejpam-4396	266	32	the	the	DET
ejpam-4396	266	33	zero	zero	NUM
ejpam-4396	266	34	of	of	ADP
ejpam-4396	266	35	the	the	DET
ejpam-4396	266	36	domain	domain	NOUN
ejpam-4396	266	37	.	.	PUNCT
ejpam-4396	267	1	since	since	SCONJ
ejpam-4396	267	2	f	f	PROPN
ejpam-4396	267	3	is	be	AUX
ejpam-4396	267	4	a	a	DET
ejpam-4396	267	5	homomorphism	homomorphism	NOUN
ejpam-4396	267	6	,	,	PUNCT
ejpam-4396	267	7	f(a)f(b	f(a)f(b	NOUN
ejpam-4396	267	8	)	)	PUNCT
ejpam-4396	267	9	=	=	SYM
ejpam-4396	267	10	f(ab	f(ab	PROPN
ejpam-4396	267	11	)	)	PUNCT
ejpam-4396	267	12	=	=	SYM
ejpam-4396	267	13	f(0	f(0	NOUN
ejpam-4396	267	14	)	)	PUNCT
ejpam-4396	267	15	=	=	SYM
ejpam-4396	267	16	f(ba	f(ba	X
ejpam-4396	267	17	)	)	PUNCT
ejpam-4396	267	18	=	=	SYM
ejpam-4396	267	19	f(b)f(a	f(b)f(a	NOUN
ejpam-4396	267	20	)	)	PUNCT
ejpam-4396	267	21	.	.	PUNCT
ejpam-4396	268	1	the	the	DET
ejpam-4396	268	2	next	next	ADJ
ejpam-4396	268	3	statement	statement	NOUN
ejpam-4396	268	4	,	,	PUNCT
ejpam-4396	268	5	theorem	theorem	VERB
ejpam-4396	268	6	10	10	NUM
ejpam-4396	268	7	,	,	PUNCT
ejpam-4396	268	8	says	say	VERB
ejpam-4396	268	9	that	that	SCONJ
ejpam-4396	268	10	an	an	DET
ejpam-4396	268	11	identity	identity	NOUN
ejpam-4396	268	12	of	of	ADP
ejpam-4396	268	13	a	a	DET
ejpam-4396	268	14	zero	zero	NUM
ejpam-4396	268	15	divisor	divisor	NOUN
ejpam-4396	268	16	is	be	AUX
ejpam-4396	268	17	also	also	ADV
ejpam-4396	268	18	a	a	DET
ejpam-4396	268	19	zero	zero	NUM
ejpam-4396	268	20	divisor	divisor	NOUN
ejpam-4396	268	21	.	.	PUNCT
ejpam-4396	269	1	theorem	theorem	VERB
ejpam-4396	269	2	10	10	NUM
ejpam-4396	269	3	.	.	PUNCT
ejpam-4396	270	1	let	let	VERB
ejpam-4396	270	2	g	g	PRON
ejpam-4396	270	3	be	be	AUX
ejpam-4396	270	4	a	a	DET
ejpam-4396	270	5	g	g	NOUN
ejpam-4396	270	6	-	-	PUNCT
ejpam-4396	270	7	group	group	NOUN
ejpam-4396	270	8	with	with	ADP
ejpam-4396	270	9	a	a	DET
ejpam-4396	270	10	zero	zero	NUM
ejpam-4396	270	11	element	element	NOUN
ejpam-4396	270	12	0	0	NUM
ejpam-4396	270	13	,	,	PUNCT
ejpam-4396	270	14	and	and	CCONJ
ejpam-4396	270	15	a	a	DET
ejpam-4396	270	16	∈	∈	NOUN
ejpam-4396	270	17	g	g	NOUN
ejpam-4396	270	18	with	with	ADP
ejpam-4396	270	19	identity	identity	NOUN
ejpam-4396	270	20	e	e	NOUN
ejpam-4396	270	21	for	for	ADP
ejpam-4396	270	22	which	which	PRON
ejpam-4396	270	23	a	a	PRON
ejpam-4396	270	24	has	have	VERB
ejpam-4396	270	25	an	an	DET
ejpam-4396	270	26	inverse	inverse	NOUN
ejpam-4396	270	27	b.	b.	NOUN
ejpam-4396	270	28	if	if	SCONJ
ejpam-4396	270	29	a	a	PRON
ejpam-4396	270	30	is	be	AUX
ejpam-4396	270	31	a	a	DET
ejpam-4396	270	32	zero	zero	NUM
ejpam-4396	270	33	divisor	divisor	NOUN
ejpam-4396	270	34	,	,	PUNCT
ejpam-4396	270	35	then	then	ADV
ejpam-4396	270	36	so	so	ADV
ejpam-4396	270	37	is	be	AUX
ejpam-4396	270	38	e.	e.	PROPN
ejpam-4396	270	39	proof	proof	PROPN
ejpam-4396	270	40	.	.	PUNCT
ejpam-4396	271	1	note	note	VERB
ejpam-4396	271	2	that	that	SCONJ
ejpam-4396	271	3	e	e	PROPN
ejpam-4396	271	4	̸=	̸=	PROPN
ejpam-4396	271	5	0	0	NUM
ejpam-4396	271	6	,	,	PUNCT
ejpam-4396	271	7	otherwise	otherwise	ADV
ejpam-4396	271	8	a	a	DET
ejpam-4396	271	9	=	=	SYM
ejpam-4396	271	10	ea	ea	NOUN
ejpam-4396	271	11	=	=	SYM
ejpam-4396	271	12	0a	0a	PROPN
ejpam-4396	271	13	=	=	SYM
ejpam-4396	271	14	0	0	PROPN
ejpam-4396	271	15	,	,	PUNCT
ejpam-4396	271	16	which	which	PRON
ejpam-4396	271	17	is	be	AUX
ejpam-4396	271	18	a	a	DET
ejpam-4396	271	19	contradiction	contradiction	NOUN
ejpam-4396	271	20	since	since	SCONJ
ejpam-4396	271	21	a	a	PRON
ejpam-4396	271	22	is	be	AUX
ejpam-4396	271	23	a	a	DET
ejpam-4396	271	24	zero	zero	NUM
ejpam-4396	271	25	divisor	divisor	NOUN
ejpam-4396	271	26	.	.	PUNCT
ejpam-4396	272	1	also	also	ADV
ejpam-4396	272	2	,	,	PUNCT
ejpam-4396	272	3	there	there	PRON
ejpam-4396	272	4	exist	exist	VERB
ejpam-4396	272	5	c	c	PROPN
ejpam-4396	272	6	∈	∈	PROPN
ejpam-4396	272	7	g	g	NOUN
ejpam-4396	272	8	with	with	ADP
ejpam-4396	272	9	c	c	PROPN
ejpam-4396	272	10	̸=	̸=	PROPN
ejpam-4396	272	11	0	0	NUM
ejpam-4396	272	12	and	and	CCONJ
ejpam-4396	272	13	ac	ac	PROPN
ejpam-4396	273	1	=	=	SYM
ejpam-4396	273	2	0	0	PUNCT
ejpam-4396	274	1	=	=	SYM
ejpam-4396	274	2	ca	ca	NOUN
ejpam-4396	274	3	.	.	PUNCT
ejpam-4396	275	1	hence	hence	ADV
ejpam-4396	275	2	,	,	PUNCT
ejpam-4396	275	3	ec	ec	PROPN
ejpam-4396	275	4	=	=	SYM
ejpam-4396	275	5	bac	bac	PROPN
ejpam-4396	275	6	=	=	SYM
ejpam-4396	275	7	b0	b0	PROPN
ejpam-4396	275	8	=	=	SYM
ejpam-4396	275	9	0	0	NUM
ejpam-4396	275	10	.	.	PUNCT
ejpam-4396	276	1	thus	thus	ADV
ejpam-4396	276	2	,	,	PUNCT
ejpam-4396	276	3	e	e	X
ejpam-4396	276	4	is	be	AUX
ejpam-4396	276	5	also	also	ADV
ejpam-4396	276	6	a	a	DET
ejpam-4396	276	7	zero	zero	NUM
ejpam-4396	276	8	divisor	divisor	NOUN
ejpam-4396	276	9	.	.	PUNCT
ejpam-4396	277	1	this	this	DET
ejpam-4396	277	2	section	section	NOUN
ejpam-4396	277	3	is	be	AUX
ejpam-4396	277	4	culminated	culminate	VERB
ejpam-4396	277	5	with	with	ADP
ejpam-4396	277	6	two	two	NUM
ejpam-4396	277	7	corollaries	corollary	NOUN
ejpam-4396	277	8	,	,	PUNCT
ejpam-4396	277	9	corollary	corollary	ADJ
ejpam-4396	277	10	4	4	NUM
ejpam-4396	277	11	and	and	CCONJ
ejpam-4396	277	12	corollary	corollary	ADJ
ejpam-4396	277	13	5	5	NUM
ejpam-4396	277	14	providing	provide	VERB
ejpam-4396	277	15	another	another	DET
ejpam-4396	277	16	way	way	NOUN
ejpam-4396	277	17	of	of	ADP
ejpam-4396	277	18	constructing	construct	VERB
ejpam-4396	277	19	a	a	DET
ejpam-4396	277	20	g	g	NOUN
ejpam-4396	277	21	-	-	PUNCT
ejpam-4396	277	22	subgroup	subgroup	NOUN
ejpam-4396	277	23	.	.	PUNCT
ejpam-4396	278	1	corollary	corollary	ADJ
ejpam-4396	278	2	4	4	NUM
ejpam-4396	278	3	.	.	PUNCT
ejpam-4396	279	1	let	let	VERB
ejpam-4396	279	2	g	g	PRON
ejpam-4396	279	3	be	be	AUX
ejpam-4396	279	4	a	a	DET
ejpam-4396	279	5	g	g	NOUN
ejpam-4396	279	6	-	-	PUNCT
ejpam-4396	279	7	group	group	NOUN
ejpam-4396	279	8	with	with	ADP
ejpam-4396	279	9	a	a	DET
ejpam-4396	279	10	zero	zero	NUM
ejpam-4396	279	11	element	element	NOUN
ejpam-4396	279	12	0	0	NUM
ejpam-4396	279	13	,	,	PUNCT
ejpam-4396	279	14	and	and	CCONJ
ejpam-4396	279	15	x	x	PUNCT
ejpam-4396	279	16	∈	∈	PROPN
ejpam-4396	279	17	g	g	NOUN
ejpam-4396	279	18	with	with	ADP
ejpam-4396	279	19	identity	identity	NOUN
ejpam-4396	279	20	e	e	NOUN
ejpam-4396	279	21	and	and	CCONJ
ejpam-4396	279	22	inverse	inverse	NOUN
ejpam-4396	279	23	y.	y.	NOUN
ejpam-4396	279	24	if	if	SCONJ
ejpam-4396	279	25	x	x	PRON
ejpam-4396	279	26	is	be	AUX
ejpam-4396	279	27	a	a	DET
ejpam-4396	279	28	unit	unit	NOUN
ejpam-4396	279	29	and	and	CCONJ
ejpam-4396	279	30	a	a	DET
ejpam-4396	279	31	zero	zero	NUM
ejpam-4396	279	32	divisor	divisor	NOUN
ejpam-4396	279	33	,	,	PUNCT
ejpam-4396	279	34	then	then	ADV
ejpam-4396	279	35	so	so	ADV
ejpam-4396	279	36	is	be	AUX
ejpam-4396	279	37	y.	y.	NOUN
ejpam-4396	279	38	proof	proof	NOUN
ejpam-4396	279	39	.	.	PUNCT
ejpam-4396	280	1	note	note	VERB
ejpam-4396	280	2	that	that	SCONJ
ejpam-4396	280	3	y	y	PROPN
ejpam-4396	280	4	̸=	̸=	PROPN
ejpam-4396	280	5	0	0	NUM
ejpam-4396	280	6	,	,	PUNCT
ejpam-4396	280	7	otherwise	otherwise	ADV
ejpam-4396	280	8	e	e	X
ejpam-4396	280	9	=	=	PUNCT
ejpam-4396	280	10	xy	xy	PROPN
ejpam-4396	281	1	=	=	PUNCT
ejpam-4396	281	2	x0	x0	PROPN
ejpam-4396	281	3	=	=	PUNCT
ejpam-4396	281	4	0	0	PROPN
ejpam-4396	281	5	,	,	PUNCT
ejpam-4396	281	6	which	which	PRON
ejpam-4396	281	7	is	be	AUX
ejpam-4396	281	8	a	a	DET
ejpam-4396	281	9	contradiction	contradiction	NOUN
ejpam-4396	281	10	to	to	PART
ejpam-4396	281	11	theorem	theorem	VERB
ejpam-4396	281	12	10	10	NUM
ejpam-4396	281	13	.	.	PUNCT
ejpam-4396	282	1	since	since	SCONJ
ejpam-4396	282	2	x	x	PRON
ejpam-4396	282	3	is	be	AUX
ejpam-4396	282	4	a	a	DET
ejpam-4396	282	5	zero	zero	NUM
ejpam-4396	282	6	divisor	divisor	NOUN
ejpam-4396	282	7	,	,	PUNCT
ejpam-4396	282	8	there	there	PRON
ejpam-4396	282	9	exist	exist	VERB
ejpam-4396	282	10	w	w	PROPN
ejpam-4396	282	11	∈	∈	PROPN
ejpam-4396	282	12	g	g	NOUN
ejpam-4396	282	13	with	with	ADP
ejpam-4396	282	14	w	w	PROPN
ejpam-4396	282	15	̸=	̸=	PROPN
ejpam-4396	282	16	0	0	NUM
ejpam-4396	282	17	and	and	CCONJ
ejpam-4396	282	18	xw	xw	PROPN
ejpam-4396	282	19	=	=	SYM
ejpam-4396	282	20	0	0	NUM
ejpam-4396	283	1	=	=	SYM
ejpam-4396	283	2	wx	wx	PROPN
ejpam-4396	283	3	.	.	PUNCT
ejpam-4396	284	1	also	also	ADV
ejpam-4396	284	2	,	,	PUNCT
ejpam-4396	284	3	since	since	SCONJ
ejpam-4396	284	4	x	x	PRON
ejpam-4396	284	5	is	be	AUX
ejpam-4396	284	6	a	a	DET
ejpam-4396	284	7	unit	unit	NOUN
ejpam-4396	284	8	,	,	PUNCT
ejpam-4396	284	9	e	e	PROPN
ejpam-4396	284	10	is	be	AUX
ejpam-4396	284	11	also	also	ADV
ejpam-4396	284	12	an	an	DET
ejpam-4396	284	13	identity	identity	NOUN
ejpam-4396	284	14	of	of	ADP
ejpam-4396	284	15	its	its	PRON
ejpam-4396	284	16	inverse	inverse	NOUN
ejpam-4396	284	17	,	,	PUNCT
ejpam-4396	284	18	y.	y.	PROPN
ejpam-4396	284	19	hence	hence	ADV
ejpam-4396	284	20	,	,	PUNCT
ejpam-4396	284	21	yw	yw	PROPN
ejpam-4396	284	22	=	=	PUNCT
ejpam-4396	284	23	yew	yew	PROPN
ejpam-4396	285	1	=	=	PUNCT
ejpam-4396	285	2	yyxw	yyxw	NOUN
ejpam-4396	285	3	=	=	PUNCT
ejpam-4396	286	1	yy0	yy0	X
ejpam-4396	286	2	=	=	SYM
ejpam-4396	286	3	0	0	PROPN
ejpam-4396	286	4	.	.	PUNCT
ejpam-4396	287	1	thus	thus	ADV
ejpam-4396	287	2	,	,	PUNCT
ejpam-4396	287	3	y	y	PROPN
ejpam-4396	287	4	is	be	AUX
ejpam-4396	287	5	also	also	ADV
ejpam-4396	287	6	a	a	DET
ejpam-4396	287	7	zero	zero	NUM
ejpam-4396	287	8	divisor	divisor	NOUN
ejpam-4396	287	9	.	.	PUNCT
ejpam-4396	288	1	corollary	corollary	ADJ
ejpam-4396	288	2	5	5	NUM
ejpam-4396	288	3	.	.	PUNCT
ejpam-4396	289	1	let	let	VERB
ejpam-4396	289	2	g	g	PRON
ejpam-4396	289	3	be	be	AUX
ejpam-4396	289	4	a	a	DET
ejpam-4396	289	5	g	g	NOUN
ejpam-4396	289	6	-	-	PUNCT
ejpam-4396	289	7	group	group	NOUN
ejpam-4396	289	8	with	with	ADP
ejpam-4396	289	9	a	a	DET
ejpam-4396	289	10	zero	zero	NUM
ejpam-4396	289	11	element	element	NOUN
ejpam-4396	289	12	.	.	PUNCT
ejpam-4396	290	1	the	the	DET
ejpam-4396	290	2	subset	subset	NOUN
ejpam-4396	290	3	d	d	X
ejpam-4396	290	4	=	=	PUNCT
ejpam-4396	290	5	{	{	PUNCT
ejpam-4396	290	6	d	d	PROPN
ejpam-4396	290	7	∈	∈	PROPN
ejpam-4396	290	8	g	g	NOUN
ejpam-4396	290	9	:	:	PUNCT
ejpam-4396	290	10	d	d	X
ejpam-4396	290	11	is	be	AUX
ejpam-4396	290	12	a	a	DET
ejpam-4396	290	13	unit	unit	NOUN
ejpam-4396	290	14	zero	zero	NUM
ejpam-4396	290	15	divisor	divisor	NOUN
ejpam-4396	290	16	}	}	PUNCT
ejpam-4396	290	17	∪	∪	X
ejpam-4396	290	18	{	{	PUNCT
ejpam-4396	290	19	0	0	NUM
ejpam-4396	290	20	}	}	PUNCT
ejpam-4396	290	21	is	be	AUX
ejpam-4396	290	22	a	a	DET
ejpam-4396	290	23	g	g	NOUN
ejpam-4396	290	24	-	-	PUNCT
ejpam-4396	290	25	subgroup	subgroup	NOUN
ejpam-4396	290	26	of	of	ADP
ejpam-4396	290	27	g.	g.	PROPN
ejpam-4396	290	28	acknowledgements	acknowledgement	NOUN
ejpam-4396	290	29	the	the	DET
ejpam-4396	290	30	authors	author	NOUN
ejpam-4396	290	31	would	would	AUX
ejpam-4396	290	32	like	like	VERB
ejpam-4396	290	33	to	to	PART
ejpam-4396	290	34	thank	thank	VERB
ejpam-4396	290	35	the	the	DET
ejpam-4396	290	36	rural	rural	ADJ
ejpam-4396	290	37	engineering	engineering	NOUN
ejpam-4396	290	38	and	and	CCONJ
ejpam-4396	290	39	technology	technology	NOUN
ejpam-4396	290	40	center	center	NOUN
ejpam-4396	290	41	of	of	ADP
ejpam-4396	290	42	negros	negros	PROPN
ejpam-4396	290	43	oriental	oriental	ADJ
ejpam-4396	290	44	state	state	PROPN
ejpam-4396	290	45	university	university	PROPN
ejpam-4396	290	46	for	for	ADP
ejpam-4396	290	47	partially	partially	ADV
ejpam-4396	290	48	supporting	support	VERB
ejpam-4396	290	49	this	this	DET
ejpam-4396	290	50	research	research	NOUN
ejpam-4396	290	51	.	.	PUNCT
ejpam-4396	291	1	references	reference	NOUN
ejpam-4396	291	2	896	896	NUM
ejpam-4396	291	3	references	reference	NOUN
ejpam-4396	291	4	[	[	X
ejpam-4396	291	5	1	1	NUM
ejpam-4396	291	6	]	]	X
ejpam-4396	291	7	j	j	PROPN
ejpam-4396	291	8	a	a	DET
ejpam-4396	291	9	caraquil	caraquil	PROPN
ejpam-4396	291	10	,	,	PUNCT
ejpam-4396	291	11	j	j	PROPN
ejpam-4396	291	12	t	t	PROPN
ejpam-4396	291	13	ubat	ubat	VERB
ejpam-4396	291	14	,	,	PUNCT
ejpam-4396	291	15	r	r	NOUN
ejpam-4396	291	16	c	c	NOUN
ejpam-4396	291	17	abrasaldo	abrasaldo	NOUN
ejpam-4396	291	18	,	,	PUNCT
ejpam-4396	291	19	and	and	CCONJ
ejpam-4396	291	20	m	m	PROPN
ejpam-4396	291	21	p	p	NOUN
ejpam-4396	291	22	baldado	baldado	NOUN
ejpam-4396	291	23	.	.	PUNCT
ejpam-4396	292	1	some	some	DET
ejpam-4396	292	2	properties	property	NOUN
ejpam-4396	292	3	of	of	ADP
ejpam-4396	292	4	the	the	DET
ejpam-4396	292	5	ubat	ubat	ADJ
ejpam-4396	292	6	-	-	PUNCT
ejpam-4396	292	7	space	space	NOUN
ejpam-4396	292	8	and	and	CCONJ
ejpam-4396	292	9	a	a	DET
ejpam-4396	292	10	related	related	ADJ
ejpam-4396	292	11	structure	structure	NOUN
ejpam-4396	292	12	.	.	PUNCT
ejpam-4396	293	1	eur	eur	PROPN
ejpam-4396	293	2	.	.	PUNCT
ejpam-4396	294	1	j.	j.	PROPN
ejpam-4396	294	2	math	math	PROPN
ejpam-4396	294	3	.	.	PUNCT
ejpam-4396	295	1	appl	appl	PROPN
ejpam-4396	295	2	,	,	PUNCT
ejpam-4396	295	3	1:1	1:1	NUM
ejpam-4396	295	4	,	,	PUNCT
ejpam-4396	295	5	2021	2021	NUM
ejpam-4396	295	6	.	.	PUNCT
ejpam-4396	296	1	[	[	X
ejpam-4396	296	2	2	2	NUM
ejpam-4396	296	3	]	]	PUNCT
ejpam-4396	296	4	f.	f.	PROPN
ejpam-4396	296	5	fatehi	fatehi	PROPN
ejpam-4396	296	6	and	and	CCONJ
ejpam-4396	296	7	m	m	PROPN
ejpam-4396	296	8	r	r	NOUN
ejpam-4396	296	9	molaei	molaei	NOUN
ejpam-4396	296	10	.	.	PUNCT
ejpam-4396	297	1	on	on	ADP
ejpam-4396	297	2	completely	completely	ADV
ejpam-4396	297	3	simple	simple	ADJ
ejpam-4396	297	4	semigroups	semigroup	NOUN
ejpam-4396	297	5	.	.	PUNCT
ejpam-4396	298	1	acta	acta	PROPN
ejpam-4396	298	2	mathematica	mathematica	PROPN
ejpam-4396	298	3	academiae	academiae	PROPN
ejpam-4396	298	4	paedagogicae	paedagogicae	VERB
ejpam-4396	298	5	nýıregyháziensis	nýıregyháziensis	NOUN
ejpam-4396	298	6	,	,	PUNCT
ejpam-4396	298	7	28:95–102	28:95–102	NUM
ejpam-4396	298	8	,	,	PUNCT
ejpam-4396	298	9	2012	2012	NUM
ejpam-4396	298	10	.	.	PUNCT
ejpam-4396	299	1	[	[	X
ejpam-4396	299	2	3	3	X
ejpam-4396	299	3	]	]	X
ejpam-4396	299	4	j	j	PROPN
ejpam-4396	299	5	b	b	PROPN
ejpam-4396	299	6	fraleigh	fraleigh	PROPN
ejpam-4396	299	7	.	.	PUNCT
ejpam-4396	300	1	a	a	DET
ejpam-4396	300	2	first	first	ADJ
ejpam-4396	300	3	course	course	NOUN
ejpam-4396	300	4	in	in	ADP
ejpam-4396	300	5	abstract	abstract	ADJ
ejpam-4396	300	6	algebra	algebra	NOUN
ejpam-4396	300	7	,	,	PUNCT
ejpam-4396	300	8	7th	7th	NOUN
ejpam-4396	300	9	,	,	PUNCT
ejpam-4396	300	10	2003	2003	NUM
ejpam-4396	300	11	.	.	PUNCT
ejpam-4396	301	1	[	[	X
ejpam-4396	301	2	4	4	X
ejpam-4396	301	3	]	]	X
ejpam-4396	301	4	j	j	PROPN
ejpam-4396	301	5	f	f	PROPN
ejpam-4396	301	6	humphreys	humphreys	PROPN
ejpam-4396	301	7	and	and	CCONJ
ejpam-4396	301	8	q	q	PROPN
ejpam-4396	301	9	liu	liu	PROPN
ejpam-4396	301	10	.	.	PUNCT
ejpam-4396	302	1	a	a	DET
ejpam-4396	302	2	course	course	NOUN
ejpam-4396	302	3	in	in	ADP
ejpam-4396	302	4	group	group	NOUN
ejpam-4396	302	5	theory	theory	NOUN
ejpam-4396	302	6	,	,	PUNCT
ejpam-4396	302	7	volume	volume	NOUN
ejpam-4396	302	8	6	6	NUM
ejpam-4396	302	9	.	.	PUNCT
ejpam-4396	303	1	oxford	oxford	PROPN
ejpam-4396	303	2	university	university	PROPN
ejpam-4396	303	3	press	press	NOUN
ejpam-4396	303	4	on	on	ADP
ejpam-4396	303	5	demand	demand	NOUN
ejpam-4396	303	6	,	,	PUNCT
ejpam-4396	303	7	1996	1996	NUM
ejpam-4396	303	8	.	.	PUNCT
ejpam-4396	304	1	[	[	X
ejpam-4396	304	2	5	5	NUM
ejpam-4396	304	3	]	]	X
ejpam-4396	304	4	i	i	PRON
ejpam-4396	304	5	kleiner	kleiner	PROPN
ejpam-4396	304	6	et	et	PROPN
ejpam-4396	304	7	al	al	PROPN
ejpam-4396	304	8	.	.	PUNCT
ejpam-4396	305	1	a	a	DET
ejpam-4396	305	2	history	history	NOUN
ejpam-4396	305	3	of	of	ADP
ejpam-4396	305	4	abstract	abstract	ADJ
ejpam-4396	305	5	algebra	algebra	NOUN
ejpam-4396	305	6	.	.	PUNCT
ejpam-4396	306	1	springer	springer	NOUN
ejpam-4396	306	2	science	science	PROPN
ejpam-4396	306	3	&	&	CCONJ
ejpam-4396	306	4	business	business	NOUN
ejpam-4396	306	5	media	medium	NOUN
ejpam-4396	306	6	,	,	PUNCT
ejpam-4396	306	7	2007	2007	NUM
ejpam-4396	306	8	.	.	PUNCT
ejpam-4396	307	1	[	[	X
ejpam-4396	307	2	6	6	NUM
ejpam-4396	307	3	]	]	PUNCT
ejpam-4396	307	4	a	a	DET
ejpam-4396	307	5	b	b	PROPN
ejpam-4396	307	6	saeid	saeid	PROPN
ejpam-4396	307	7	,	,	PUNCT
ejpam-4396	307	8	a	a	DET
ejpam-4396	307	9	rezaei	rezaei	NOUN
ejpam-4396	307	10	,	,	PUNCT
ejpam-4396	307	11	and	and	CCONJ
ejpam-4396	307	12	a	a	DET
ejpam-4396	307	13	radfar	radfar	NOUN
ejpam-4396	307	14	.	.	PUNCT
ejpam-4396	308	1	a	a	DET
ejpam-4396	308	2	generalization	generalization	NOUN
ejpam-4396	308	3	of	of	ADP
ejpam-4396	308	4	groups	group	NOUN
ejpam-4396	308	5	.	.	PUNCT
ejpam-4396	309	1	atti	atti	PROPN
ejpam-4396	309	2	della	della	PROPN
ejpam-4396	309	3	accademia	accademia	PROPN
ejpam-4396	309	4	peloritana	peloritana	PROPN
ejpam-4396	309	5	dei	dei	ADP
ejpam-4396	309	6	pericolanti	pericolanti	ADJ
ejpam-4396	309	7	-	-	ADJ
ejpam-4396	309	8	classe	classe	ADJ
ejpam-4396	309	9	di	di	PROPN
ejpam-4396	309	10	scienze	scienze	PROPN
ejpam-4396	309	11	fisiche	fisiche	PROPN
ejpam-4396	309	12	,	,	PUNCT
ejpam-4396	309	13	matematiche	matematiche	PROPN
ejpam-4396	309	14	e	e	X
ejpam-4396	309	15	naturali	naturali	PROPN
ejpam-4396	309	16	,	,	PUNCT
ejpam-4396	309	17	96(1):4	96(1):4	NOUN
ejpam-4396	309	18	,	,	PUNCT
ejpam-4396	309	19	2018	2018	NUM
ejpam-4396	309	20	.	.	PUNCT
ejpam-4396	310	1	[	[	X
ejpam-4396	310	2	7	7	X
ejpam-4396	310	3	]	]	SYM
ejpam-4396	310	4	m	m	VERB
ejpam-4396	310	5	r	r	NOUN
ejpam-4396	310	6	a	a	DET
ejpam-4396	310	7	zand	zand	PROPN
ejpam-4396	310	8	and	and	CCONJ
ejpam-4396	310	9	s	s	PROPN
ejpam-4396	310	10	rostami	rostami	NOUN
ejpam-4396	310	11	.	.	PUNCT
ejpam-4396	311	1	some	some	DET
ejpam-4396	311	2	topological	topological	ADJ
ejpam-4396	311	3	aspects	aspect	NOUN
ejpam-4396	311	4	of	of	ADP
ejpam-4396	311	5	generalized	generalized	ADJ
ejpam-4396	311	6	groups	group	NOUN
ejpam-4396	311	7	and	and	CCONJ
ejpam-4396	311	8	pseudonorms	pseudonorm	NOUN
ejpam-4396	311	9	on	on	ADP
ejpam-4396	311	10	them	they	PRON
ejpam-4396	311	11	.	.	PUNCT
ejpam-4396	312	1	honam	honam	PROPN
ejpam-4396	312	2	mathematical	mathematical	PROPN
ejpam-4396	312	3	journal	journal	PROPN
ejpam-4396	312	4	,	,	PUNCT
ejpam-4396	312	5	40(4):661–669	40(4):661–669	PROPN
ejpam-4396	312	6	,	,	PUNCT
ejpam-4396	312	7	2018	2018	NUM
ejpam-4396	312	8	.	.	PUNCT