id	sid	tid	token	lemma	pos
m-782	1	1	ijo	ijo	PROPN
m-782	1	2	international	international	PROPN
m-782	1	3	journal	journal	PROPN
m-782	1	4	of	of	ADP
m-782	1	5	mathematics	mathematics	PROPN
m-782	1	6	(	(	PUNCT
m-782	1	7	issn	issn	PROPN
m-782	1	8	:	:	PUNCT
m-782	1	9	2992	2992	NUM
m-782	1	10	-	-	SYM
m-782	1	11	4421	4421	NUM
m-782	1	12	)	)	PUNCT
m-782	2	1	michael	michael	PROPN
m-782	2	2	n.	n.	PROPN
m-782	2	3	john	john	PROPN
m-782	2	4	*	*	PROPN
m-782	2	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-782	2	6	volume	volume	NOUN
m-782	2	7	07	07	NUM
m-782	2	8	issue	issue	NOUN
m-782	2	9	01	01	NUM
m-782	2	10	||	||	NUM
m-782	2	11	january	january	PROPN
m-782	2	12	.	.	PUNCT
m-782	2	13	,	,	PUNCT
m-782	2	14	2024	2024	NUM
m-782	2	15	||	||	X
m-782	2	16	para-	para-	NUM
m-782	2	17	�	�	PROPN
m-782	2	18	relations	relation	NOUN
m-782	2	19	and	and	CCONJ
m-782	2	20	hirsch	hirsch	PROPN
m-782	2	21	length	length	NOUN
m-782	2	22	in	in	ADP
m-782	2	23	residually	residually	ADV
m-782	2	24	nilpotent	nilpotent	ADJ
m-782	2	25	groups	group	NOUN
m-782	2	26	michael	michael	PROPN
m-782	2	27	n.	n.	PROPN
m-782	2	28	john	john	PROPN
m-782	2	29	department	department	PROPN
m-782	2	30	of	of	ADP
m-782	2	31	mathematics	mathematics	PROPN
m-782	2	32	,	,	PUNCT
m-782	2	33	akwa	akwa	ADJ
m-782	2	34	ibom	ibom	ADJ
m-782	2	35	state	state	NOUN
m-782	2	36	university	university	PROPN
m-782	2	37	,	,	PUNCT
m-782	2	38	nigeria	nigeria	PROPN
m-782	2	39	stephen	stephen	PROPN
m-782	2	40	i.	i.	PROPN
m-782	2	41	okeke	okeke	PROPN
m-782	2	42	department	department	PROPN
m-782	2	43	of	of	ADP
m-782	2	44	industrial	industrial	ADJ
m-782	2	45	mathematics	mathematic	NOUN
m-782	2	46	and	and	CCONJ
m-782	2	47	applied	applied	ADJ
m-782	2	48	statistics	statistic	NOUN
m-782	2	49	,	,	PUNCT
m-782	2	50	david	david	PROPN
m-782	2	51	umahi	umahi	PROPN
m-782	2	52	federal	federal	PROPN
m-782	2	53	university	university	PROPN
m-782	2	54	of	of	ADP
m-782	2	55	health	health	PROPN
m-782	2	56	sciences	sciences	PROPN
m-782	2	57	,	,	PUNCT
m-782	2	58	uburu	uburu	PROPN
m-782	2	59	,	,	PUNCT
m-782	2	60	ebonyi	ebonyi	PROPN
m-782	2	61	state	state	PROPN
m-782	2	62	,	,	PUNCT
m-782	2	63	nigeria	nigeria	PROPN
m-782	2	64	.	.	PUNCT
m-782	3	1	boniface	boniface	PROPN
m-782	3	2	o.	o.	PROPN
m-782	3	3	nwala	nwala	PROPN
m-782	3	4	department	department	PROPN
m-782	3	5	of	of	ADP
m-782	3	6	mathematics	mathematics	PROPN
m-782	3	7	,	,	PUNCT
m-782	3	8	ignatius	ignatius	PROPN
m-782	3	9	ajuru	ajuru	PROPN
m-782	3	10	university	university	PROPN
m-782	3	11	of	of	ADP
m-782	3	12	education	education	NOUN
m-782	3	13	,	,	PUNCT
m-782	3	14	rumuolumeni	rumuolumeni	NOUN
m-782	3	15	,	,	PUNCT
m-782	3	16	port	port	NOUN
m-782	3	17	harcourt	harcourt	PROPN
m-782	3	18	,	,	PUNCT
m-782	3	19	rivers	river	NOUN
m-782	3	20	state	state	PROPN
m-782	3	21	,	,	PUNCT
m-782	3	22	nigeria	nigeria	PROPN
m-782	3	23	udoaka	udoaka	ADV
m-782	3	24	otobong	otobong	PROPN
m-782	3	25	.	.	PUNCT
m-782	4	1	g.	g.	PROPN
m-782	4	2	department	department	PROPN
m-782	4	3	of	of	ADP
m-782	4	4	mathematics	mathematic	NOUN
m-782	4	5	,	,	PUNCT
m-782	4	6	akwa	akwa	ADJ
m-782	4	7	ibom	ibom	ADJ
m-782	4	8	state	state	NOUN
m-782	4	9	university	university	PROPN
m-782	4	10	,	,	PUNCT
m-782	4	11	nigeria	nigeria	PROPN
m-782	4	12	abstract	abstract	ADV
m-782	4	13	this	this	DET
m-782	4	14	research	research	NOUN
m-782	4	15	explores	explore	VERB
m-782	4	16	the	the	DET
m-782	4	17	interplay	interplay	NOUN
m-782	4	18	between	between	ADP
m-782	4	19	residually	residually	ADV
m-782	4	20	nilpotent	nilpotent	ADJ
m-782	4	21	groups	group	NOUN
m-782	4	22	�	�	PROPN
m-782	4	23	and	and	CCONJ
m-782	4	24	�	�	PROPN
m-782	4	25	,	,	PUNCT
m-782	4	26	focusing	focus	VERB
m-782	4	27	on	on	ADP
m-782	4	28	their	their	PRON
m-782	4	29	relationship	relationship	NOUN
m-782	4	30	through	through	ADP
m-782	4	31	the	the	DET
m-782	4	32	lens	lens	NOUN
m-782	4	33	of	of	ADP
m-782	4	34	para-	para-	NOUN
m-782	4	35	�	�	PROPN
m-782	4	36	conditions	condition	NOUN
m-782	4	37	and	and	CCONJ
m-782	4	38	the	the	DET
m-782	4	39	hirsch	hirsch	PROPN
m-782	4	40	length	length	PROPN
m-782	4	41	.	.	PUNCT
m-782	5	1	we	we	PRON
m-782	5	2	establish	establish	VERB
m-782	5	3	criteria	criterion	NOUN
m-782	5	4	for	for	SCONJ
m-782	5	5	�	�	PROPN
m-782	5	6	to	to	PART
m-782	5	7	be	be	AUX
m-782	5	8	para-	para-	VERB
m-782	5	9	�	�	PROPN
m-782	5	10	concerning	concern	VERB
m-782	5	11	monomorphisms	monomorphism	NOUN
m-782	5	12	inducing	induce	VERB
m-782	5	13	isomorphisms	isomorphism	NOUN
m-782	5	14	between	between	ADP
m-782	5	15	corresponding	correspond	VERB
m-782	5	16	lower	low	ADJ
m-782	5	17	central	central	ADJ
m-782	5	18	quotients	quotient	NOUN
m-782	5	19	of	of	ADP
m-782	5	20	�	�	PROPN
m-782	5	21	and	and	CCONJ
m-782	5	22	�	�	PROPN
m-782	5	23	.	.	PUNCT
m-782	6	1	specifically	specifically	ADV
m-782	6	2	,	,	PUNCT
m-782	6	3	we	we	PRON
m-782	6	4	investigate	investigate	VERB
m-782	6	5	these	these	DET
m-782	6	6	conditions	condition	NOUN
m-782	6	7	in	in	ADP
m-782	6	8	the	the	DET
m-782	6	9	context	context	NOUN
m-782	6	10	of	of	ADP
m-782	6	11	finitely	finitely	ADV
m-782	6	12	generated	generate	VERB
m-782	6	13	residually	residually	ADV
m-782	6	14	nilpotent	nilpotent	ADJ
m-782	6	15	groups	group	NOUN
m-782	6	16	.	.	PUNCT
m-782	7	1	further	far	ADV
m-782	7	2	,	,	PUNCT
m-782	7	3	for	for	ADP
m-782	7	4	certain	certain	ADJ
m-782	7	5	polycyclic	polycyclic	ADJ
m-782	7	6	groups	group	NOUN
m-782	7	7	,	,	PUNCT
m-782	7	8	we	we	PRON
m-782	7	9	establish	establish	VERB
m-782	7	10	connections	connection	NOUN
m-782	7	11	between	between	ADP
m-782	7	12	para-	para-	NUM
m-782	7	13	�	�	PROPN
m-782	7	14	relations	relation	NOUN
m-782	7	15	and	and	CCONJ
m-782	7	16	the	the	DET
m-782	7	17	equality	equality	NOUN
m-782	7	18	of	of	ADP
m-782	7	19	hirsch	hirsch	PROPN
m-782	7	20	lengths	length	NOUN
m-782	7	21	.	.	PUNCT
m-782	8	1	additionally	additionally	ADV
m-782	8	2	,	,	PUNCT
m-782	8	3	we	we	PRON
m-782	8	4	delve	delve	VERB
m-782	8	5	into	into	ADP
m-782	8	6	the	the	DET
m-782	8	7	pro	pro	ADJ
m-782	8	8	-	-	ADJ
m-782	8	9	nilpotent	nilpotent	ADJ
m-782	8	10	completions	completion	NOUN
m-782	8	11	of	of	ADP
m-782	8	12	these	these	DET
m-782	8	13	polycyclic	polycyclic	ADJ
m-782	8	14	groups	group	NOUN
m-782	8	15	,	,	PUNCT
m-782	8	16	demonstrating	demonstrate	VERB
m-782	8	17	their	their	PRON
m-782	8	18	local	local	ADJ
m-782	8	19	polycyclic	polycyclic	NOUN
m-782	8	20	nature	nature	NOUN
m-782	8	21	.	.	PUNCT
m-782	9	1	keywords	keyword	NOUN
m-782	9	2	:	:	PUNCT
m-782	9	3	residually	residually	ADV
m-782	9	4	nilpotent	nilpotent	ADJ
m-782	9	5	groups	group	NOUN
m-782	9	6	,	,	PUNCT
m-782	9	7	para-	para-	NUM
m-782	9	8	�	�	PROPN
m-782	9	9	relations	relation	NOUN
m-782	9	10	,	,	PUNCT
m-782	9	11	hirsch	hirsch	PROPN
m-782	9	12	length	length	PROPN
m-782	9	13	,	,	PUNCT
m-782	9	14	lower	low	ADJ
m-782	9	15	central	central	ADJ
m-782	9	16	quotients	quotient	NOUN
m-782	9	17	,	,	PUNCT
m-782	9	18	pro	pro	ADJ
m-782	9	19	-	-	ADJ
m-782	9	20	nilpotent	nilpotent	ADJ
m-782	9	21	completions	completion	NOUN
m-782	9	22	,	,	PUNCT
m-782	9	23	polycyclic	polycyclic	NOUN
m-782	9	24	groups	group	NOUN
m-782	9	25	.	.	PUNCT
m-782	10	1	doi	doi	PROPN
m-782	10	2	10.5281	10.5281	NUM
m-782	10	3	/	/	SYM
m-782	10	4	zenodo.10511744	zenodo.10511744	PROPN
m-782	10	5	ijo	ijo	PROPN
m-782	10	6	journals	journal	NOUN
m-782	10	7	volume	volume	NOUN
m-782	10	8	07	07	NUM
m-782	10	9	|	|	NOUN
m-782	10	10	issue	issue	NOUN
m-782	10	11	01	01	NUM
m-782	11	1	|	|	CCONJ
m-782	11	2	january	january	PROPN
m-782	11	3	2024	2024	NUM
m-782	11	4	|	|	ADV
m-782	11	5	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-782	11	6	1	1	NUM
m-782	11	7	ijo	ijo	PROPN
m-782	11	8	international	international	PROPN
m-782	11	9	journal	journal	PROPN
m-782	11	10	of	of	ADP
m-782	11	11	mathematics	mathematics	PROPN
m-782	11	12	(	(	PUNCT
m-782	11	13	issn	issn	PROPN
m-782	11	14	:	:	PUNCT
m-782	11	15	2992	2992	NUM
m-782	11	16	-	-	SYM
m-782	11	17	4421	4421	NUM
m-782	11	18	)	)	PUNCT
m-782	12	1	michael	michael	PROPN
m-782	12	2	n.	n.	PROPN
m-782	12	3	john	john	PROPN
m-782	12	4	*	*	PROPN
m-782	12	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-782	12	6	volume	volume	NOUN
m-782	12	7	07	07	NUM
m-782	12	8	issue	issue	NOUN
m-782	12	9	01	01	NUM
m-782	12	10	||	||	NUM
m-782	12	11	january	january	PROPN
m-782	12	12	.	.	PUNCT
m-782	12	13	,	,	PUNCT
m-782	12	14	2024	2024	NUM
m-782	12	15	||	||	NOUN
m-782	12	16	1	1	NUM
m-782	12	17	.	.	X
m-782	12	18	introduction	introduction	NOUN
m-782	12	19	residually	residually	ADV
m-782	12	20	nilpotent	nilpotent	ADJ
m-782	12	21	groups	group	NOUN
m-782	12	22	play	play	VERB
m-782	12	23	a	a	DET
m-782	12	24	pivotal	pivotal	ADJ
m-782	12	25	role	role	NOUN
m-782	12	26	in	in	ADP
m-782	12	27	group	group	NOUN
m-782	12	28	theory	theory	NOUN
m-782	12	29	,	,	PUNCT
m-782	12	30	and	and	CCONJ
m-782	12	31	understanding	understand	VERB
m-782	12	32	their	their	PRON
m-782	12	33	relationships	relationship	NOUN
m-782	12	34	is	be	AUX
m-782	12	35	essential	essential	ADJ
m-782	12	36	for	for	ADP
m-782	12	37	exploring	explore	VERB
m-782	12	38	the	the	DET
m-782	12	39	underlying	underlie	VERB
m-782	12	40	algebraic	algebraic	ADJ
m-782	12	41	structures	structure	NOUN
m-782	12	42	.	.	PUNCT
m-782	13	1	the	the	DET
m-782	13	2	study	study	NOUN
m-782	13	3	by	by	ADP
m-782	13	4	[	[	X
m-782	13	5	1	1	NUM
m-782	13	6	]	]	PUNCT
m-782	13	7	provides	provide	VERB
m-782	13	8	foundational	foundational	ADJ
m-782	13	9	insights	insight	NOUN
m-782	13	10	into	into	ADP
m-782	13	11	para-	para-	NOUN
m-782	13	12	�	�	PROPN
m-782	13	13	conditions	condition	NOUN
m-782	13	14	in	in	ADP
m-782	13	15	group	group	NOUN
m-782	13	16	theory	theory	NOUN
m-782	13	17	,	,	PUNCT
m-782	13	18	particularly	particularly	ADV
m-782	13	19	in	in	ADP
m-782	13	20	the	the	DET
m-782	13	21	context	context	NOUN
m-782	13	22	of	of	ADP
m-782	13	23	residually	residually	ADV
m-782	13	24	nilpotent	nilpotent	ADJ
m-782	13	25	groups	group	NOUN
m-782	13	26	.	.	PUNCT
m-782	14	1	hall	hall	PROPN
m-782	14	2	's	's	PART
m-782	14	3	work	work	NOUN
m-782	14	4	lays	lay	VERB
m-782	14	5	the	the	DET
m-782	14	6	groundwork	groundwork	NOUN
m-782	14	7	for	for	ADP
m-782	14	8	understanding	understand	VERB
m-782	14	9	the	the	DET
m-782	14	10	interconnections	interconnection	NOUN
m-782	14	11	between	between	ADP
m-782	14	12	groups	group	NOUN
m-782	14	13	and	and	CCONJ
m-782	14	14	the	the	DET
m-782	14	15	criteria	criterion	NOUN
m-782	14	16	for	for	ADP
m-782	14	17	para-	para-	NOUN
m-782	14	18	�	�	PROPN
m-782	14	19	relations.the	relations.the	DET
m-782	14	20	concept	concept	NOUN
m-782	14	21	of	of	ADP
m-782	14	22	hirsch	hirsch	PROPN
m-782	14	23	length	length	PROPN
m-782	14	24	has	have	AUX
m-782	14	25	been	be	AUX
m-782	14	26	extensively	extensively	ADV
m-782	14	27	explored	explore	VERB
m-782	14	28	in	in	ADP
m-782	14	29	relation	relation	NOUN
m-782	14	30	to	to	ADP
m-782	14	31	finitely	finitely	ADV
m-782	14	32	generated	generate	VERB
m-782	14	33	groups	group	NOUN
m-782	14	34	.	.	PUNCT
m-782	15	1	[	[	X
m-782	15	2	2	2	NUM
m-782	15	3	]	]	VERB
m-782	15	4	's	's	PART
m-782	15	5	seminal	seminal	ADJ
m-782	15	6	work	work	NOUN
m-782	15	7	(	(	PUNCT
m-782	15	8	1967	1967	NUM
m-782	15	9	)	)	PUNCT
m-782	15	10	investigates	investigate	VERB
m-782	15	11	the	the	DET
m-782	15	12	properties	property	NOUN
m-782	15	13	of	of	ADP
m-782	15	14	the	the	DET
m-782	15	15	hirsch	hirsch	PROPN
m-782	15	16	length	length	NOUN
m-782	15	17	and	and	CCONJ
m-782	15	18	its	its	PRON
m-782	15	19	implications	implication	NOUN
m-782	15	20	in	in	ADP
m-782	15	21	the	the	DET
m-782	15	22	study	study	NOUN
m-782	15	23	of	of	ADP
m-782	15	24	groups.the	groups.the	DET
m-782	15	25	exploration	exploration	NOUN
m-782	15	26	of	of	ADP
m-782	15	27	para-	para-	NOUN
m-782	15	28	�	�	PROPN
m-782	15	29	relations	relation	NOUN
m-782	15	30	within	within	ADP
m-782	15	31	polycyclic	polycyclic	NOUN
m-782	15	32	groups	group	NOUN
m-782	15	33	is	be	AUX
m-782	15	34	addressed	address	VERB
m-782	15	35	by	by	ADP
m-782	15	36	[	[	X
m-782	15	37	3	3	NUM
m-782	15	38	]	]	PUNCT
m-782	15	39	.	.	PUNCT
m-782	16	1	this	this	DET
m-782	16	2	work	work	NOUN
m-782	16	3	delves	delve	VERB
m-782	16	4	into	into	ADP
m-782	16	5	the	the	DET
m-782	16	6	specific	specific	ADJ
m-782	16	7	conditions	condition	NOUN
m-782	16	8	and	and	CCONJ
m-782	16	9	implications	implication	NOUN
m-782	16	10	of	of	ADP
m-782	16	11	para-	para-	NOUN
m-782	16	12	�	�	PROPN
m-782	16	13	relations	relation	NOUN
m-782	16	14	in	in	ADP
m-782	16	15	the	the	DET
m-782	16	16	context	context	NOUN
m-782	16	17	of	of	ADP
m-782	16	18	polycyclic	polycyclic	NOUN
m-782	16	19	structures.the	structures.the	DET
m-782	16	20	study	study	NOUN
m-782	16	21	of	of	ADP
m-782	16	22	pro	pro	ADJ
m-782	16	23	-	-	ADJ
m-782	16	24	nilpotent	nilpotent	ADJ
m-782	16	25	completions	completion	NOUN
m-782	16	26	in	in	ADP
m-782	16	27	the	the	DET
m-782	16	28	realm	realm	NOUN
m-782	16	29	of	of	ADP
m-782	16	30	polycyclic	polycyclic	NOUN
m-782	16	31	groups	group	NOUN
m-782	16	32	is	be	AUX
m-782	16	33	discussed	discuss	VERB
m-782	16	34	by	by	ADP
m-782	16	35	[	[	X
m-782	16	36	4	4	NUM
m-782	16	37	]	]	PUNCT
m-782	16	38	and	and	CCONJ
m-782	16	39	itprovides	itprovide	VERB
m-782	16	40	insights	insight	NOUN
m-782	16	41	into	into	ADP
m-782	16	42	the	the	DET
m-782	16	43	local	local	ADJ
m-782	16	44	polycyclic	polycyclic	NOUN
m-782	16	45	nature	nature	NOUN
m-782	16	46	of	of	ADP
m-782	16	47	these	these	DET
m-782	16	48	completions	completion	NOUN
m-782	16	49	,	,	PUNCT
m-782	16	50	contributing	contribute	VERB
m-782	16	51	to	to	ADP
m-782	16	52	the	the	DET
m-782	16	53	broader	broad	ADJ
m-782	16	54	understanding	understanding	NOUN
m-782	16	55	of	of	ADP
m-782	16	56	their	their	PRON
m-782	16	57	properties	property	NOUN
m-782	16	58	.	.	PUNCT
m-782	17	1	this	this	DET
m-782	17	2	research	research	NOUN
m-782	17	3	focuses	focus	VERB
m-782	17	4	on	on	ADP
m-782	17	5	establishing	establish	VERB
m-782	17	6	and	and	CCONJ
m-782	17	7	characterizing	characterize	VERB
m-782	17	8	para-	para-	NUM
m-782	17	9	�	�	NOUN
m-782	17	10	relations	relation	NOUN
m-782	17	11	between	between	ADP
m-782	17	12	residually	residually	ADV
m-782	17	13	nilpotent	nilpotent	ADJ
m-782	17	14	groups	group	NOUN
m-782	17	15	�	�	PROPN
m-782	17	16	and	and	CCONJ
m-782	17	17	�	�	PROPN
m-782	17	18	,	,	PUNCT
m-782	17	19	with	with	ADP
m-782	17	20	a	a	DET
m-782	17	21	particular	particular	ADJ
m-782	17	22	emphasis	emphasis	NOUN
m-782	17	23	on	on	ADP
m-782	17	24	monomorphisms	monomorphism	NOUN
m-782	17	25	inducing	induce	VERB
m-782	17	26	isomorphisms	isomorphism	NOUN
m-782	17	27	between	between	ADP
m-782	17	28	their	their	PRON
m-782	17	29	lower	low	ADJ
m-782	17	30	central	central	ADJ
m-782	17	31	quotients	quotient	NOUN
m-782	17	32	.	.	PUNCT
m-782	18	1	we	we	PRON
m-782	18	2	extend	extend	VERB
m-782	18	3	our	our	PRON
m-782	18	4	investigation	investigation	NOUN
m-782	18	5	to	to	ADP
m-782	18	6	finitely	finitely	ADV
m-782	18	7	generated	generate	VERB
m-782	18	8	groups	group	NOUN
m-782	18	9	and	and	CCONJ
m-782	18	10	explore	explore	VERB
m-782	18	11	conditions	condition	NOUN
m-782	18	12	for	for	SCONJ
m-782	18	13	�	�	PROPN
m-782	18	14	to	to	PART
m-782	18	15	be	be	AUX
m-782	18	16	para-	para-	VERB
m-782	18	17	�	�	PROPN
m-782	18	18	.	.	PUNCT
m-782	19	1	moreover	moreover	ADV
m-782	19	2	,	,	PUNCT
m-782	19	3	we	we	PRON
m-782	19	4	explore	explore	VERB
m-782	19	5	the	the	DET
m-782	19	6	implications	implication	NOUN
m-782	19	7	of	of	ADP
m-782	19	8	para-	para-	NOUN
m-782	19	9	�	�	PROPN
m-782	19	10	relations	relation	NOUN
m-782	19	11	on	on	ADP
m-782	19	12	the	the	DET
m-782	19	13	hirsch	hirsch	PROPN
m-782	19	14	length	length	NOUN
m-782	19	15	of	of	ADP
m-782	19	16	certain	certain	ADJ
m-782	19	17	polycyclic	polycyclic	NOUN
m-782	19	18	groups	group	NOUN
m-782	19	19	.	.	PUNCT
m-782	20	1	2	2	X
m-782	20	2	.	.	X
m-782	20	3	preliminary	preliminary	ADJ
m-782	20	4	definition	definition	NOUN
m-782	20	5	(	(	PUNCT
m-782	20	6	residually	residually	ADV
m-782	20	7	nilpotent	nilpotent	ADJ
m-782	20	8	groups	group	NOUN
m-782	20	9	)	)	PUNCT
m-782	20	10	2.1	2.1	NUM
m-782	20	11	.	.	PUNCT
m-782	21	1	a	a	DET
m-782	21	2	group	group	NOUN
m-782	21	3	g	g	NOUN
m-782	21	4	is	be	AUX
m-782	21	5	said	say	VERB
m-782	21	6	to	to	PART
m-782	21	7	be	be	AUX
m-782	21	8	residually	residually	ADV
m-782	21	9	nilpotent	nilpotent	ADJ
m-782	21	10	if	if	SCONJ
m-782	21	11	,	,	PUNCT
m-782	21	12	for	for	ADP
m-782	21	13	every	every	DET
m-782	21	14	non	non	ADJ
m-782	21	15	-	-	ADJ
m-782	21	16	identity	identity	ADJ
m-782	21	17	element	element	NOUN
m-782	21	18	g	g	NOUN
m-782	21	19	in	in	ADP
m-782	21	20	g	g	PROPN
m-782	21	21	,	,	PUNCT
m-782	21	22	there	there	PRON
m-782	21	23	exists	exist	VERB
m-782	21	24	a	a	DET
m-782	21	25	normal	normal	ADJ
m-782	21	26	subgroup	subgroup	NOUN
m-782	21	27	n	n	PROPN
m-782	21	28	of	of	ADP
m-782	21	29	finite	finite	ADJ
m-782	21	30	index	index	NOUN
m-782	21	31	such	such	ADJ
m-782	21	32	that	that	SCONJ
m-782	21	33	n	n	PRON
m-782	21	34	is	be	AUX
m-782	21	35	a	a	DET
m-782	21	36	nilpotent	nilpotent	ADJ
m-782	21	37	group	group	NOUN
m-782	21	38	.	.	PUNCT
m-782	22	1	in	in	ADP
m-782	22	2	other	other	ADJ
m-782	22	3	words	word	NOUN
m-782	22	4	,	,	PUNCT
m-782	22	5	every	every	DET
m-782	22	6	nonidentity	nonidentity	NOUN
m-782	22	7	element	element	NOUN
m-782	22	8	of	of	ADP
m-782	22	9	the	the	DET
m-782	22	10	group	group	NOUN
m-782	22	11	can	can	AUX
m-782	22	12	be	be	AUX
m-782	22	13	separated	separate	VERB
m-782	22	14	from	from	ADP
m-782	22	15	the	the	DET
m-782	22	16	identity	identity	NOUN
m-782	22	17	by	by	ADP
m-782	22	18	a	a	DET
m-782	22	19	finite	finite	ADJ
m-782	22	20	-	-	ADJ
m-782	22	21	index	index	NOUN
m-782	22	22	normal	normal	ADJ
m-782	22	23	subgroup	subgroup	NOUN
m-782	22	24	that	that	PRON
m-782	22	25	is	be	AUX
m-782	22	26	nilpotent	nilpotent	ADJ
m-782	22	27	.	.	PUNCT
m-782	23	1	doi	doi	NOUN
m-782	23	2	10.5281	10.5281	NUM
m-782	23	3	/	/	SYM
m-782	23	4	zenodo.10511744	zenodo.10511744	PROPN
m-782	23	5	ijo	ijo	PROPN
m-782	23	6	journals	journal	NOUN
m-782	23	7	volume	volume	NOUN
m-782	23	8	07	07	NUM
m-782	24	1	|	|	NOUN
m-782	24	2	issue	issue	NOUN
m-782	24	3	01	01	NUM
m-782	25	1	|	|	CCONJ
m-782	25	2	january	january	PROPN
m-782	25	3	2024	2024	NUM
m-782	25	4	|	|	ADV
m-782	25	5	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-782	25	6	2	2	NUM
m-782	25	7	ijo	ijo	PROPN
m-782	25	8	international	international	ADJ
m-782	25	9	journal	journal	PROPN
m-782	25	10	of	of	ADP
m-782	25	11	mathematics	mathematics	PROPN
m-782	25	12	(	(	PUNCT
m-782	25	13	issn	issn	PROPN
m-782	25	14	:	:	PUNCT
m-782	25	15	2992	2992	NUM
m-782	25	16	-	-	SYM
m-782	25	17	4421	4421	NUM
m-782	25	18	)	)	PUNCT
m-782	26	1	michael	michael	PROPN
m-782	26	2	n.	n.	PROPN
m-782	26	3	john	john	PROPN
m-782	26	4	*	*	PROPN
m-782	26	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-782	26	6	volume	volume	NOUN
m-782	26	7	07	07	NUM
m-782	26	8	issue	issue	NOUN
m-782	26	9	01	01	NUM
m-782	26	10	||	||	NUM
m-782	26	11	january	january	PROPN
m-782	26	12	.	.	PUNCT
m-782	26	13	,	,	PUNCT
m-782	26	14	2024	2024	NUM
m-782	26	15	||	||	NOUN
m-782	26	16	example	example	NOUN
m-782	26	17	(	(	PUNCT
m-782	26	18	residually	residually	ADV
m-782	26	19	nilpotent	nilpotent	ADJ
m-782	26	20	groups	group	NOUN
m-782	26	21	)	)	PUNCT
m-782	26	22	2.2	2.2	NUM
m-782	26	23	.	.	PUNCT
m-782	27	1	consider	consider	VERB
m-782	27	2	the	the	DET
m-782	27	3	group	group	NOUN
m-782	27	4	g	g	PROPN
m-782	27	5	=	=	PROPN
m-782	27	6	z×s3	z×s3	NOUN
m-782	27	7	,	,	PUNCT
m-782	27	8	where	where	SCONJ
m-782	27	9	z	z	NOUN
m-782	27	10	is	be	AUX
m-782	27	11	the	the	DET
m-782	27	12	additive	additive	ADJ
m-782	27	13	group	group	NOUN
m-782	27	14	of	of	ADP
m-782	27	15	integers	integer	NOUN
m-782	27	16	and	and	CCONJ
m-782	27	17	s3	s3	PROPN
m-782	27	18	is	be	AUX
m-782	27	19	the	the	DET
m-782	27	20	symmetric	symmetric	ADJ
m-782	27	21	group	group	NOUN
m-782	27	22	on	on	ADP
m-782	27	23	three	three	NUM
m-782	27	24	elements	element	NOUN
m-782	27	25	.	.	PUNCT
m-782	28	1	this	this	DET
m-782	28	2	group	group	NOUN
m-782	28	3	is	be	AUX
m-782	28	4	a	a	DET
m-782	28	5	direct	direct	ADJ
m-782	28	6	product	product	NOUN
m-782	28	7	of	of	ADP
m-782	28	8	an	an	DET
m-782	28	9	infinite	infinite	ADJ
m-782	28	10	cyclic	cyclic	NOUN
m-782	28	11	group	group	NOUN
m-782	28	12	(	(	PUNCT
m-782	28	13	z	z	NOUN
m-782	28	14	)	)	PUNCT
m-782	28	15	and	and	CCONJ
m-782	28	16	a	a	DET
m-782	28	17	finite	finite	ADJ
m-782	28	18	group	group	NOUN
m-782	28	19	(	(	PUNCT
m-782	28	20	s3	s3	PROPN
m-782	28	21	)	)	PUNCT
m-782	28	22	.	.	PUNCT
m-782	29	1	the	the	DET
m-782	29	2	group	group	NOUN
m-782	29	3	g	g	PROPN
m-782	29	4	is	be	AUX
m-782	29	5	residually	residually	ADV
m-782	29	6	nilpotent	nilpotent	ADJ
m-782	29	7	because	because	SCONJ
m-782	29	8	:	:	PUNCT
m-782	29	9	1	1	X
m-782	29	10	.	.	X
m-782	29	11	for	for	ADP
m-782	29	12	any	any	DET
m-782	29	13	non	non	ADJ
m-782	29	14	-	-	ADJ
m-782	29	15	identity	identity	ADJ
m-782	29	16	element	element	NOUN
m-782	29	17	(	(	PUNCT
m-782	29	18	n	n	CCONJ
m-782	29	19	,	,	PUNCT
m-782	29	20	e	e	NOUN
m-782	29	21	)	)	PUNCT
m-782	29	22	∈	∈	PROPN
m-782	29	23	g	g	NOUN
m-782	29	24	,	,	PUNCT
m-782	29	25	where	where	SCONJ
m-782	29	26	n	n	PRON
m-782	29	27	is	be	AUX
m-782	29	28	a	a	DET
m-782	29	29	non	non	ADJ
m-782	29	30	-	-	ADJ
m-782	29	31	zero	zero	ADJ
m-782	29	32	integer	integer	NOUN
m-782	29	33	and	and	CCONJ
m-782	29	34	e	e	NOUN
m-782	29	35	is	be	AUX
m-782	29	36	the	the	DET
m-782	29	37	identity	identity	NOUN
m-782	29	38	element	element	NOUN
m-782	29	39	of	of	ADP
m-782	29	40	s3	s3	PROPN
m-782	29	41	,	,	PUNCT
m-782	29	42	we	we	PRON
m-782	29	43	can	can	AUX
m-782	29	44	consider	consider	VERB
m-782	29	45	the	the	DET
m-782	29	46	subgroup	subgroup	NOUN
m-782	29	47	n	n	NOUN
m-782	29	48	=	=	PRON
m-782	29	49	{	{	PUNCT
m-782	29	50	(	(	PUNCT
m-782	29	51	0,e	0,e	NOUN
m-782	29	52	)	)	PUNCT
m-782	29	53	}	}	PUNCT
m-782	29	54	.	.	PUNCT
m-782	30	1	this	this	DET
m-782	30	2	subgroup	subgroup	NOUN
m-782	30	3	is	be	AUX
m-782	30	4	of	of	ADP
m-782	30	5	finite	finite	ADJ
m-782	30	6	index	index	NOUN
m-782	30	7	,	,	PUNCT
m-782	30	8	and	and	CCONJ
m-782	30	9	n	n	PROPN
m-782	30	10	is	be	AUX
m-782	30	11	nilpotent	nilpotent	ADJ
m-782	30	12	.	.	PUNCT
m-782	31	1	2	2	X
m-782	31	2	.	.	X
m-782	31	3	for	for	ADP
m-782	31	4	any	any	DET
m-782	31	5	non	non	ADJ
m-782	31	6	-	-	ADJ
m-782	31	7	identity	identity	ADJ
m-782	31	8	element	element	NOUN
m-782	31	9	(	(	PUNCT
m-782	31	10	0,σ	0,σ	PROPN
m-782	31	11	)	)	PUNCT
m-782	31	12	∈	∈	PROPN
m-782	31	13	g	g	NOUN
m-782	31	14	,	,	PUNCT
m-782	31	15	where	where	SCONJ
m-782	31	16	σ	σ	PROPN
m-782	31	17	is	be	AUX
m-782	31	18	a	a	DET
m-782	31	19	non	non	ADJ
m-782	31	20	-	-	ADJ
m-782	31	21	identity	identity	ADJ
m-782	31	22	permutation	permutation	NOUN
m-782	31	23	in	in	ADP
m-782	31	24	s3	s3	PROPN
m-782	31	25	,	,	PUNCT
m-782	31	26	we	we	PRON
m-782	31	27	can	can	AUX
m-782	31	28	consider	consider	VERB
m-782	31	29	the	the	DET
m-782	31	30	subgroup	subgroup	NOUN
m-782	31	31	n	n	NOUN
m-782	31	32	=	=	PRON
m-782	31	33	{	{	PUNCT
m-782	31	34	(	(	PUNCT
m-782	31	35	0,σ	0,σ	NOUN
m-782	31	36	)	)	PUNCT
m-782	31	37	,	,	PUNCT
m-782	31	38	(	(	PUNCT
m-782	31	39	0,e	0,e	NOUN
m-782	31	40	)	)	PUNCT
m-782	31	41	}	}	PUNCT
m-782	31	42	.	.	PUNCT
m-782	32	1	this	this	DET
m-782	32	2	subgroup	subgroup	NOUN
m-782	32	3	is	be	AUX
m-782	32	4	of	of	ADP
m-782	32	5	finite	finite	ADJ
m-782	32	6	index	index	NOUN
m-782	32	7	,	,	PUNCT
m-782	32	8	and	and	CCONJ
m-782	32	9	n	n	PROPN
m-782	32	10	is	be	AUX
m-782	32	11	nilpotent	nilpotent	ADJ
m-782	32	12	.	.	PUNCT
m-782	33	1	thus	thus	ADV
m-782	33	2	,	,	PUNCT
m-782	33	3	g	g	PROPN
m-782	33	4	=	=	SYM
m-782	33	5	z	z	NOUN
m-782	33	6	×	×	PROPN
m-782	33	7	s3	s3	PROPN
m-782	33	8	is	be	AUX
m-782	33	9	an	an	DET
m-782	33	10	example	example	NOUN
m-782	33	11	of	of	ADP
m-782	33	12	a	a	DET
m-782	33	13	residually	residually	ADV
m-782	33	14	nilpotent	nilpotent	ADJ
m-782	33	15	group	group	NOUN
m-782	33	16	definition	definition	NOUN
m-782	33	17	(	(	PUNCT
m-782	33	18	para-	para-	NUM
m-782	33	19	�	�	PROPN
m-782	33	20	relations	relation	NOUN
m-782	33	21	)	)	PUNCT
m-782	33	22	2.3	2.3	NUM
m-782	33	23	.	.	PUNCT
m-782	34	1	let	let	VERB
m-782	34	2	g	g	NOUN
m-782	34	3	and	and	CCONJ
m-782	34	4	h	h	NOUN
m-782	34	5	be	be	VERB
m-782	34	6	two	two	NUM
m-782	34	7	groups	group	NOUN
m-782	34	8	.	.	PUNCT
m-782	35	1	the	the	DET
m-782	35	2	relation	relation	NOUN
m-782	35	3	φ	φ	NOUN
m-782	35	4	:	:	PUNCT
m-782	35	5	g→h	g→h	PROPN
m-782	35	6	is	be	AUX
m-782	35	7	a	a	DET
m-782	35	8	para	para	NOUN
m-782	35	9	-	-	PUNCT
m-782	35	10	g	g	NOUN
m-782	35	11	relation	relation	NOUN
m-782	35	12	if	if	SCONJ
m-782	35	13	,	,	PUNCT
m-782	35	14	for	for	ADP
m-782	35	15	every	every	DET
m-782	35	16	normal	normal	ADJ
m-782	35	17	subgroup	subgroup	NOUN
m-782	35	18	n	n	PROPN
m-782	35	19	of	of	ADP
m-782	35	20	g	g	PROPN
m-782	35	21	,	,	PUNCT
m-782	35	22	the	the	DET
m-782	35	23	induced	induce	VERB
m-782	35	24	homomorphism	homomorphism	NOUN
m-782	35	25	φn	φn	ADP
m-782	35	26	:	:	PUNCT
m-782	35	27	g	g	PROPN
m-782	35	28	/	/	SYM
m-782	35	29	n→h	n→h	NOUN
m-782	35	30	/	/	SYM
m-782	35	31	φ(n	φ(n	NOUN
m-782	35	32	)	)	PUNCT
m-782	35	33	is	be	AUX
m-782	35	34	an	an	DET
m-782	35	35	isomorphism	isomorphism	NOUN
m-782	35	36	,	,	PUNCT
m-782	35	37	where	where	SCONJ
m-782	35	38	φ(n	φ(n	VERB
m-782	35	39	)	)	PUNCT
m-782	35	40	=	=	PRON
m-782	35	41	{	{	PUNCT
m-782	35	42	φ(g)|g∈n	φ(g)|g∈n	PROPN
m-782	35	43	}	}	PUNCT
m-782	35	44	is	be	AUX
m-782	35	45	the	the	DET
m-782	35	46	image	image	NOUN
m-782	35	47	of	of	ADP
m-782	35	48	n	n	PROPN
m-782	35	49	under	under	ADP
m-782	35	50	φ	φ	NUM
m-782	35	51	.	.	PUNCT
m-782	36	1	in	in	ADP
m-782	36	2	simpler	simple	ADJ
m-782	36	3	terms	term	NOUN
m-782	36	4	,	,	PUNCT
m-782	36	5	a	a	DET
m-782	36	6	para	para	NOUN
m-782	36	7	-	-	PUNCT
m-782	36	8	g	g	NOUN
m-782	36	9	relation	relation	NOUN
m-782	36	10	is	be	AUX
m-782	36	11	a	a	DET
m-782	36	12	condition	condition	NOUN
m-782	36	13	on	on	ADP
m-782	36	14	a	a	DET
m-782	36	15	group	group	NOUN
m-782	36	16	homomorphism	homomorphism	NOUN
m-782	36	17	φ	φ	NOUN
m-782	36	18	:	:	PUNCT
m-782	36	19	g→h	g→h	NOUN
m-782	36	20	such	such	ADJ
m-782	36	21	that	that	SCONJ
m-782	36	22	the	the	DET
m-782	36	23	homomorphism	homomorphism	NOUN
m-782	36	24	induces	induce	VERB
m-782	36	25	isomorphisms	isomorphism	NOUN
m-782	36	26	between	between	ADP
m-782	36	27	corresponding	correspond	VERB
m-782	36	28	lower	low	ADJ
m-782	36	29	central	central	ADJ
m-782	36	30	quotients	quotient	NOUN
m-782	36	31	for	for	ADP
m-782	36	32	every	every	DET
m-782	36	33	normal	normal	ADJ
m-782	36	34	subgroup	subgroup	NOUN
m-782	36	35	of	of	ADP
m-782	36	36	g.	g.	PROPN
m-782	36	37	for	for	ADP
m-782	36	38	a	a	DET
m-782	36	39	good	good	ADJ
m-782	36	40	homomorphism	homomorphism	NOUN
m-782	36	41	and	and	CCONJ
m-782	36	42	the	the	DET
m-782	36	43	generators	generator	NOUN
m-782	36	44	of	of	ADP
m-782	36	45	its	its	PRON
m-782	36	46	inner	inner	ADJ
m-782	36	47	automorphism	automorphism	NOUN
m-782	36	48	see	see	VERB
m-782	36	49	[	[	X
m-782	36	50	29	29	NUM
m-782	36	51	]	]	PUNCT
m-782	36	52	and	and	CCONJ
m-782	36	53	[	[	X
m-782	36	54	30	30	NUM
m-782	36	55	]	]	PUNCT
m-782	36	56	.	.	PUNCT
m-782	37	1	doi	doi	PROPN
m-782	37	2	10.5281	10.5281	NUM
m-782	37	3	/	/	SYM
m-782	37	4	zenodo.10511744	zenodo.10511744	PROPN
m-782	37	5	ijo	ijo	PROPN
m-782	37	6	journals	journal	NOUN
m-782	37	7	volume	volume	NOUN
m-782	37	8	07	07	NUM
m-782	38	1	|	|	NOUN
m-782	38	2	issue	issue	NOUN
m-782	38	3	01	01	NUM
m-782	39	1	|	|	CCONJ
m-782	39	2	january	january	PROPN
m-782	39	3	2024	2024	NUM
m-782	39	4	|	|	ADV
m-782	39	5	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-782	39	6	3	3	NUM
m-782	39	7	ijo	ijo	PROPN
m-782	39	8	international	international	PROPN
m-782	39	9	journal	journal	PROPN
m-782	39	10	of	of	ADP
m-782	39	11	mathematics	mathematics	PROPN
m-782	39	12	(	(	PUNCT
m-782	39	13	issn	issn	PROPN
m-782	39	14	:	:	PUNCT
m-782	39	15	2992	2992	NUM
m-782	39	16	-	-	SYM
m-782	39	17	4421	4421	NUM
m-782	39	18	)	)	PUNCT
m-782	40	1	michael	michael	PROPN
m-782	40	2	n.	n.	PROPN
m-782	40	3	john	john	PROPN
m-782	40	4	*	*	PROPN
m-782	40	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-782	40	6	volume	volume	NOUN
m-782	40	7	07	07	NUM
m-782	40	8	issue	issue	NOUN
m-782	40	9	01	01	NUM
m-782	40	10	||	||	NUM
m-782	40	11	january	january	PROPN
m-782	40	12	.	.	PUNCT
m-782	40	13	,	,	PUNCT
m-782	40	14	2024	2024	NUM
m-782	40	15	||	||	NOUN
m-782	40	16	example	example	NOUN
m-782	40	17	(	(	PUNCT
m-782	40	18	para-	para-	NUM
m-782	40	19	�	�	PROPN
m-782	40	20	relations	relation	NOUN
m-782	40	21	)	)	PUNCT
m-782	40	22	2.4	2.4	NUM
m-782	40	23	.	.	PUNCT
m-782	41	1	let	let	VERB
m-782	41	2	's	us	PRON
m-782	41	3	consider	consider	VERB
m-782	41	4	two	two	NUM
m-782	41	5	groups	group	NOUN
m-782	41	6	g	g	NOUN
m-782	41	7	and	and	CCONJ
m-782	41	8	h	h	NOUN
m-782	41	9	with	with	ADP
m-782	41	10	the	the	DET
m-782	41	11	following	follow	VERB
m-782	41	12	properties	property	NOUN
m-782	41	13	:	:	PUNCT
m-782	41	14	g	g	PROPN
m-782	41	15	=	=	SYM
m-782	41	16	⟨a	⟨a	PROPN
m-782	41	17	,	,	PUNCT
m-782	41	18	b|	b|	PROPN
m-782	41	19	a2	a2	PROPN
m-782	41	20	=	=	PUNCT
m-782	41	21	b2	b2	PROPN
m-782	41	22	=	=	SYM
m-782	41	23	(	(	PUNCT
m-782	41	24	ab)2	ab)2	PROPN
m-782	41	25	=	=	PUNCT
m-782	41	26	e⟩	e⟩	NOUN
m-782	41	27	h	h	NOUN
m-782	41	28	=	=	PUNCT
m-782	41	29	⟨x	⟨x	NUM
m-782	41	30	,	,	PUNCT
m-782	42	1	y|	y|	NOUN
m-782	42	2	x2	x2	NOUN
m-782	42	3	=	=	PUNCT
m-782	42	4	y2	y2	NOUN
m-782	42	5	=	=	SYM
m-782	43	1	(	(	PUNCT
m-782	43	2	xy)3	xy)3	PROPN
m-782	43	3	=	=	PRON
m-782	43	4	e⟩	e⟩	NOUN
m-782	43	5	define	define	VERB
m-782	43	6	a	a	DET
m-782	43	7	group	group	NOUN
m-782	43	8	homomorphism	homomorphism	NOUN
m-782	43	9	φ	φ	NOUN
m-782	43	10	:	:	PUNCT
m-782	43	11	g→h	g→h	NOUN
m-782	43	12	by	by	ADP
m-782	43	13	mapping	map	VERB
m-782	43	14	a	a	PRON
m-782	43	15	to	to	ADP
m-782	43	16	x	x	PROPN
m-782	43	17	and	and	CCONJ
m-782	43	18	b	b	NOUN
m-782	43	19	to	to	PART
m-782	43	20	y.	y.	VERB
m-782	43	21	this	this	PRON
m-782	43	22	homomorphism	homomorphism	NOUN
m-782	43	23	φ	φ	PROPN
m-782	43	24	is	be	AUX
m-782	43	25	a	a	DET
m-782	43	26	para	para	NOUN
m-782	43	27	-	-	PUNCT
m-782	43	28	g	g	NOUN
m-782	43	29	relation	relation	NOUN
m-782	43	30	if	if	SCONJ
m-782	43	31	,	,	PUNCT
m-782	43	32	for	for	ADP
m-782	43	33	every	every	DET
m-782	43	34	normal	normal	ADJ
m-782	43	35	subgroup	subgroup	NOUN
m-782	43	36	n	n	PROPN
m-782	43	37	of	of	ADP
m-782	43	38	g	g	PROPN
m-782	43	39	,	,	PUNCT
m-782	43	40	the	the	DET
m-782	43	41	induced	induce	VERB
m-782	43	42	homomorphism	homomorphism	NOUN
m-782	43	43	φn	φn	ADP
m-782	43	44	:	:	PUNCT
m-782	43	45	g	g	PROPN
m-782	43	46	/	/	SYM
m-782	43	47	n→h	n→h	NOUN
m-782	43	48	/	/	SYM
m-782	43	49	φ(n	φ(n	NOUN
m-782	43	50	)	)	PUNCT
m-782	43	51	is	be	AUX
m-782	43	52	an	an	DET
m-782	43	53	isomorphism	isomorphism	NOUN
m-782	43	54	.	.	PUNCT
m-782	44	1	for	for	ADP
m-782	44	2	example	example	NOUN
m-782	44	3	,	,	PUNCT
m-782	44	4	consider	consider	VERB
m-782	44	5	the	the	DET
m-782	44	6	normal	normal	ADJ
m-782	44	7	subgroup	subgroup	NOUN
m-782	44	8	n	n	NOUN
m-782	44	9	=	=	SYM
m-782	44	10	⟨a⟩	⟨a⟩	PROPN
m-782	44	11	of	of	ADP
m-782	44	12	g.	g.	PROPN
m-782	44	13	the	the	DET
m-782	44	14	induced	induce	VERB
m-782	44	15	homomorphism	homomorphism	PROPN
m-782	44	16	φn	φn	ADJ
m-782	44	17	:	:	PUNCT
m-782	44	18	g	g	PROPN
m-782	44	19	/	/	SYM
m-782	44	20	n→h	n→h	NOUN
m-782	44	21	/	/	SYM
m-782	44	22	φ(n	φ(n	NOUN
m-782	44	23	)	)	PUNCT
m-782	44	24	is	be	AUX
m-782	44	25	an	an	DET
m-782	44	26	isomorphism	isomorphism	NOUN
m-782	44	27	because	because	SCONJ
m-782	44	28	:	:	PUNCT
m-782	44	29	φn(en	φn(en	X
m-782	44	30	)	)	PUNCT
m-782	44	31	=	=	PUNCT
m-782	44	32	φ(e	φ(e	X
m-782	44	33	)	)	PUNCT
m-782	44	34	=	=	SYM
m-782	44	35	e	e	NOUN
m-782	44	36	=	=	PUNCT
m-782	44	37	φ(n	φ(n	NOUN
m-782	44	38	)	)	PUNCT
m-782	44	39	φn(bn	φn(bn	NOUN
m-782	44	40	)	)	PUNCT
m-782	44	41	=	=	PUNCT
m-782	45	1	φ(b	φ(b	X
m-782	45	2	)	)	PUNCT
m-782	45	3	=	=	SYM
m-782	46	1	y	y	NOUN
m-782	46	2	=	=	PUNCT
m-782	46	3	φ(n	φ(n	PROPN
m-782	46	4	)	)	PUNCT
m-782	46	5	this	this	PRON
m-782	46	6	holds	hold	VERB
m-782	46	7	for	for	ADP
m-782	46	8	every	every	DET
m-782	46	9	normal	normal	ADJ
m-782	46	10	subgroup	subgroup	NOUN
m-782	46	11	of	of	ADP
m-782	46	12	g	g	PROPN
m-782	46	13	,	,	PUNCT
m-782	46	14	and	and	CCONJ
m-782	46	15	therefore	therefore	ADV
m-782	46	16	,	,	PUNCT
m-782	46	17	the	the	DET
m-782	46	18	homomorphism	homomorphism	PROPN
m-782	46	19	φ	φ	PROPN
m-782	46	20	is	be	AUX
m-782	46	21	a	a	DET
m-782	46	22	para	para	NOUN
m-782	46	23	-	-	PUNCT
m-782	46	24	g	g	NOUN
m-782	46	25	relation	relation	NOUN
m-782	46	26	between	between	ADP
m-782	46	27	g	g	PROPN
m-782	46	28	and	and	CCONJ
m-782	46	29	h.	h.	PROPN
m-782	46	30	definition	definition	NOUN
m-782	46	31	(	(	PUNCT
m-782	46	32	hirsch	hirsch	PROPN
m-782	46	33	length	length	PROPN
m-782	46	34	)	)	PUNCT
m-782	46	35	2.5	2.5	NUM
m-782	46	36	.	.	PUNCT
m-782	47	1	the	the	DET
m-782	47	2	hirsch	hirsch	PROPN
m-782	47	3	length	length	NOUN
m-782	47	4	of	of	ADP
m-782	47	5	a	a	DET
m-782	47	6	group	group	NOUN
m-782	47	7	g	g	NOUN
m-782	47	8	,	,	PUNCT
m-782	47	9	denoted	denote	VERB
m-782	47	10	as	as	ADP
m-782	47	11	h(g	h(g	NOUN
m-782	47	12	)	)	PUNCT
m-782	47	13	,	,	PUNCT
m-782	47	14	is	be	AUX
m-782	47	15	a	a	DET
m-782	47	16	non	non	ADJ
m-782	47	17	-	-	ADJ
m-782	47	18	negative	negative	ADJ
m-782	47	19	integer	integer	NOUN
m-782	47	20	that	that	PRON
m-782	47	21	measures	measure	VERB
m-782	47	22	the	the	DET
m-782	47	23	growth	growth	NOUN
m-782	47	24	rate	rate	NOUN
m-782	47	25	of	of	ADP
m-782	47	26	the	the	DET
m-782	47	27	lower	low	ADJ
m-782	47	28	central	central	ADJ
m-782	47	29	series	series	NOUN
m-782	47	30	of	of	ADP
m-782	47	31	g.	g.	PROPN
m-782	47	32	specifically	specifically	ADV
m-782	47	33	,	,	PUNCT
m-782	47	34	h(g	h(g	NOUN
m-782	47	35	)	)	PUNCT
m-782	47	36	is	be	AUX
m-782	47	37	the	the	DET
m-782	47	38	length	length	NOUN
m-782	47	39	of	of	ADP
m-782	47	40	the	the	DET
m-782	47	41	shortest	short	ADJ
m-782	47	42	possible	possible	ADJ
m-782	47	43	generating	generating	NOUN
m-782	47	44	tuple	tuple	NOUN
m-782	47	45	(	(	PUNCT
m-782	47	46	g1,g2,	g1,g2,	PROPN
m-782	47	47	…	…	SYM
m-782	47	48	,gk	,gk	PUNCT
m-782	47	49	)	)	PUNCT
m-782	47	50	for	for	ADP
m-782	47	51	g	g	PROPN
m-782	47	52	such	such	ADJ
m-782	47	53	that	that	SCONJ
m-782	47	54	the	the	DET
m-782	47	55	i	i	PROPN
m-782	47	56	-	-	PUNCT
m-782	47	57	th	th	PROPN
m-782	47	58	term	term	NOUN
m-782	47	59	of	of	ADP
m-782	47	60	the	the	DET
m-782	47	61	lower	low	ADJ
m-782	47	62	central	central	ADJ
m-782	47	63	series	series	NOUN
m-782	47	64	of	of	ADP
m-782	47	65	g	g	PROPN
m-782	47	66	is	be	AUX
m-782	47	67	generated	generate	VERB
m-782	47	68	by	by	ADP
m-782	47	69	g1,g2,	g1,g2,	PROPN
m-782	47	70	…	…	PUNCT
m-782	47	71	,gi	,gi	PUNCT
m-782	47	72	for	for	ADP
m-782	47	73	each	each	DET
m-782	47	74	i	i	PRON
m-782	47	75	from	from	ADP
m-782	47	76	1	1	NUM
m-782	47	77	to	to	ADP
m-782	47	78	k.	k.	PROPN
m-782	47	79	in	in	ADP
m-782	47	80	other	other	ADJ
m-782	47	81	words	word	NOUN
m-782	47	82	,	,	PUNCT
m-782	47	83	h(g	h(g	NOUN
m-782	47	84	)	)	PUNCT
m-782	47	85	is	be	AUX
m-782	47	86	the	the	DET
m-782	47	87	smallest	small	ADJ
m-782	47	88	integer	integer	NOUN
m-782	47	89	k	k	PROPN
m-782	47	90	such	such	ADJ
m-782	47	91	that	that	PRON
m-782	47	92	g(k)={e	g(k)={e	PROPN
m-782	47	93	}	}	PUNCT
m-782	47	94	,	,	PUNCT
m-782	47	95	where	where	SCONJ
m-782	47	96	g(k	g(k	NOUN
m-782	47	97	)	)	PUNCT
m-782	47	98	denotes	denote	VERB
m-782	47	99	the	the	DET
m-782	47	100	k	k	PROPN
m-782	47	101	-	-	PUNCT
m-782	47	102	th	th	VERB
m-782	47	103	term	term	NOUN
m-782	47	104	of	of	ADP
m-782	47	105	the	the	DET
m-782	47	106	lower	low	ADJ
m-782	47	107	central	central	ADJ
m-782	47	108	series	series	NOUN
m-782	47	109	of	of	ADP
m-782	47	110	g.	g.	PROPN
m-782	47	111	doi	doi	PROPN
m-782	47	112	10.5281	10.5281	NUM
m-782	47	113	/	/	SYM
m-782	47	114	zenodo.10511744	zenodo.10511744	PROPN
m-782	47	115	ijo	ijo	PROPN
m-782	47	116	journals	journal	NOUN
m-782	47	117	volume	volume	NOUN
m-782	47	118	07	07	NUM
m-782	47	119	|	|	NOUN
m-782	47	120	issue	issue	NOUN
m-782	48	1	01	01	NUM
m-782	49	1	|	|	CCONJ
m-782	49	2	january	january	PROPN
m-782	49	3	2024	2024	NUM
m-782	49	4	|	|	ADV
m-782	49	5	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-782	49	6	4	4	NUM
m-782	49	7	ijo	ijo	PROPN
m-782	49	8	international	international	ADJ
m-782	49	9	journal	journal	PROPN
m-782	49	10	of	of	ADP
m-782	49	11	mathematics	mathematics	PROPN
m-782	49	12	(	(	PUNCT
m-782	49	13	issn	issn	PROPN
m-782	49	14	:	:	PUNCT
m-782	49	15	2992	2992	NUM
m-782	49	16	-	-	SYM
m-782	49	17	4421	4421	NUM
m-782	49	18	)	)	PUNCT
m-782	50	1	michael	michael	PROPN
m-782	50	2	n.	n.	PROPN
m-782	50	3	john	john	PROPN
m-782	50	4	*	*	PROPN
m-782	50	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-782	50	6	volume	volume	NOUN
m-782	50	7	07	07	NUM
m-782	50	8	issue	issue	NOUN
m-782	50	9	01	01	NUM
m-782	50	10	||	||	NUM
m-782	50	11	january	january	PROPN
m-782	50	12	.	.	PUNCT
m-782	50	13	,	,	PUNCT
m-782	50	14	2024	2024	NUM
m-782	50	15	||	||	NOUN
m-782	50	16	example	example	NOUN
m-782	50	17	(	(	PUNCT
m-782	50	18	hirsch	hirsch	PROPN
m-782	50	19	length	length	PROPN
m-782	50	20	)	)	PUNCT
m-782	50	21	2.6	2.6	NUM
m-782	50	22	.	.	PUNCT
m-782	51	1	consider	consider	VERB
m-782	51	2	the	the	DET
m-782	51	3	free	free	ADJ
m-782	51	4	group	group	NOUN
m-782	51	5	f2	f2	PROPN
m-782	51	6	on	on	ADP
m-782	51	7	two	two	NUM
m-782	51	8	generators	generator	NOUN
m-782	51	9	a	a	PRON
m-782	51	10	and	and	CCONJ
m-782	51	11	b	b	NOUN
m-782	51	12	,	,	PUNCT
m-782	51	13	i.e.	i.e.	X
m-782	51	14	,	,	PUNCT
m-782	51	15	f2=⟨a	f2=⟨a	ADJ
m-782	51	16	,	,	PUNCT
m-782	51	17	b|⟩.	b|⟩.	VERB
m-782	51	18	the	the	DET
m-782	51	19	lower	low	ADJ
m-782	51	20	central	central	ADJ
m-782	51	21	series	series	NOUN
m-782	51	22	of	of	ADP
m-782	51	23	f2	f2	PROPN
m-782	51	24	is	be	AUX
m-782	51	25	given	give	VERB
m-782	51	26	by	by	ADP
m-782	51	27	:	:	PUNCT
m-782	51	28	f2	f2	PROPN
m-782	51	29	(	(	PUNCT
m-782	51	30	1)=f2	1)=f2	NOUN
m-782	51	31	f2	f2	PROPN
m-782	51	32	(	(	PUNCT
m-782	51	33	2	2	NUM
m-782	51	34	)	)	PUNCT
m-782	51	35	=	=	NOUN
m-782	52	1	[	[	X
m-782	52	2	f2,f2	f2,f2	X
m-782	52	3	]	]	X
m-782	52	4	=	=	PUNCT
m-782	52	5	⟨[a	⟨[a	NOUN
m-782	52	6	,	,	PUNCT
m-782	52	7	b]⟩	b]⟩	ADJ
m-782	52	8	f2	f2	PROPN
m-782	52	9	(	(	PUNCT
m-782	52	10	3	3	NUM
m-782	52	11	)	)	PUNCT
m-782	52	12	=	=	NOUN
m-782	53	1	[	[	X
m-782	53	2	f2,f2	f2,f2	PROPN
m-782	53	3	(	(	PUNCT
m-782	53	4	2	2	NUM
m-782	53	5	)	)	PUNCT
m-782	53	6	]	]	PUNCT
m-782	53	7	and	and	CCONJ
m-782	53	8	so	so	ADV
m-782	53	9	on	on	ADV
m-782	53	10	.	.	PUNCT
m-782	54	1	in	in	ADP
m-782	54	2	this	this	DET
m-782	54	3	case	case	NOUN
m-782	54	4	,	,	PUNCT
m-782	54	5	the	the	DET
m-782	54	6	hirsch	hirsch	PROPN
m-782	54	7	length	length	PROPN
m-782	54	8	h(f2	h(f2	PROPN
m-782	54	9	)	)	PUNCT
m-782	54	10	is	be	AUX
m-782	54	11	2	2	NUM
m-782	54	12	because	because	SCONJ
m-782	54	13	the	the	DET
m-782	54	14	shortest	short	ADJ
m-782	54	15	generating	generate	VERB
m-782	54	16	tuple	tuple	NOUN
m-782	54	17	(	(	PUNCT
m-782	54	18	g1,g2	g1,g2	PROPN
m-782	54	19	)	)	PUNCT
m-782	54	20	is	be	AUX
m-782	54	21	(	(	PUNCT
m-782	54	22	a,[a	a,[a	PROPN
m-782	54	23	,	,	PUNCT
m-782	54	24	b	b	NOUN
m-782	54	25	]	]	X
m-782	54	26	)	)	PUNCT
m-782	54	27	,	,	PUNCT
m-782	54	28	and	and	CCONJ
m-782	54	29	f2	f2	PROPN
m-782	54	30	(	(	PUNCT
m-782	54	31	2	2	NUM
m-782	54	32	)	)	PUNCT
m-782	54	33	=	=	SYM
m-782	54	34	⟨[a	⟨[a	NOUN
m-782	54	35	,	,	PUNCT
m-782	54	36	b]⟩	b]⟩	ADJ
m-782	54	37	is	be	AUX
m-782	54	38	generated	generate	VERB
m-782	54	39	by	by	ADP
m-782	54	40	a	a	DET
m-782	54	41	and	and	CCONJ
m-782	54	42	[	[	X
m-782	54	43	a	a	X
m-782	54	44	,	,	PUNCT
m-782	54	45	b	b	NOUN
m-782	54	46	]	]	X
m-782	54	47	.	.	PUNCT
m-782	55	1	if	if	SCONJ
m-782	55	2	one	one	PRON
m-782	55	3	tries	try	VERB
m-782	55	4	to	to	PART
m-782	55	5	generate	generate	VERB
m-782	55	6	f2	f2	PROPN
m-782	55	7	(	(	PUNCT
m-782	55	8	3	3	NUM
m-782	55	9	)	)	PUNCT
m-782	55	10	,	,	PUNCT
m-782	55	11	a	a	DET
m-782	55	12	longer	long	ADJ
m-782	55	13	tuple	tuple	NOUN
m-782	55	14	is	be	AUX
m-782	55	15	needed	need	VERB
m-782	55	16	.	.	PUNCT
m-782	56	1	so	so	ADV
m-782	56	2	,	,	PUNCT
m-782	56	3	for	for	ADP
m-782	56	4	the	the	DET
m-782	56	5	free	free	ADJ
m-782	56	6	group	group	NOUN
m-782	56	7	f2,h(f2	f2,h(f2	PROPN
m-782	56	8	)	)	PUNCT
m-782	56	9	=	=	SYM
m-782	56	10	2	2	X
m-782	56	11	.	.	X
m-782	56	12	definition	definition	NOUN
m-782	56	13	(	(	PUNCT
m-782	56	14	pro	pro	ADJ
m-782	56	15	-	-	ADJ
m-782	56	16	nilpotent	nilpotent	ADJ
m-782	56	17	completions	completion	NOUN
m-782	56	18	)	)	PUNCT
m-782	56	19	2.7	2.7	NUM
m-782	56	20	.	.	PUNCT
m-782	57	1	let	let	VERB
m-782	57	2	g	g	PRON
m-782	57	3	be	be	AUX
m-782	57	4	a	a	DET
m-782	57	5	group	group	NOUN
m-782	57	6	.	.	PUNCT
m-782	58	1	the	the	DET
m-782	58	2	pronilpotent	pronilpotent	PROPN
m-782	58	3	completion	completion	NOUN
m-782	58	4	of	of	ADP
m-782	58	5	g	g	NOUN
m-782	58	6	,	,	PUNCT
m-782	58	7	denoted	denote	VERB
m-782	58	8	as	as	ADP
m-782	58	9	�	�	PROPN
m-782	58	10	�	�	PROPN
m-782	58	11	nil	nil	NOUN
m-782	58	12	or	or	CCONJ
m-782	58	13	�	�	PROPN
m-782	58	14	�	�	PROPN
m-782	58	15	�	�	PROPN
m-782	58	16	�	�	PROPN
m-782	58	17	,	,	PUNCT
m-782	58	18	is	be	AUX
m-782	58	19	the	the	DET
m-782	58	20	completion	completion	NOUN
m-782	58	21	of	of	ADP
m-782	58	22	g	g	NOUN
m-782	58	23	with	with	ADP
m-782	58	24	respect	respect	NOUN
m-782	58	25	to	to	ADP
m-782	58	26	the	the	DET
m-782	58	27	pro	pro	ADJ
m-782	58	28	-	-	ADJ
m-782	58	29	nilpotent	nilpotent	ADJ
m-782	58	30	topology	topology	NOUN
m-782	58	31	.	.	PUNCT
m-782	59	1	the	the	DET
m-782	59	2	pro	pro	ADJ
m-782	59	3	-	-	ADJ
m-782	59	4	nilpotent	nilpotent	ADJ
m-782	59	5	topology	topology	NOUN
m-782	59	6	on	on	ADP
m-782	59	7	g	g	PROPN
m-782	59	8	is	be	AUX
m-782	59	9	defined	define	VERB
m-782	59	10	by	by	ADP
m-782	59	11	the	the	DET
m-782	59	12	collection	collection	NOUN
m-782	59	13	of	of	ADP
m-782	59	14	all	all	DET
m-782	59	15	normal	normal	ADJ
m-782	59	16	subgroups	subgroup	NOUN
m-782	59	17	n	n	CCONJ
m-782	59	18	of	of	ADP
m-782	59	19	g	g	NOUN
m-782	59	20	such	such	ADJ
m-782	59	21	that	that	SCONJ
m-782	59	22	the	the	DET
m-782	59	23	quotient	quotient	NOUN
m-782	59	24	g	g	NOUN
m-782	59	25	/	/	SYM
m-782	59	26	n	n	PROPN
m-782	59	27	is	be	AUX
m-782	59	28	nilpotent	nilpotent	ADJ
m-782	59	29	.	.	PUNCT
m-782	60	1	the	the	DET
m-782	60	2	pro	pro	ADJ
m-782	60	3	-	-	ADJ
m-782	60	4	nilpotent	nilpotent	ADJ
m-782	60	5	completion	completion	NOUN
m-782	60	6	�	�	PROPN
m-782	60	7	�	�	PROPN
m-782	60	8	nil	nil	NOUN
m-782	60	9	is	be	AUX
m-782	60	10	the	the	DET
m-782	60	11	projective	projective	ADJ
m-782	60	12	limit	limit	NOUN
m-782	60	13	of	of	ADP
m-782	60	14	the	the	DET
m-782	60	15	nilpotent	nilpotent	ADJ
m-782	60	16	quotients	quotient	VERB
m-782	60	17	g	g	NOUN
m-782	60	18	/	/	SYM
m-782	60	19	n	n	NOUN
m-782	60	20	over	over	ADP
m-782	60	21	all	all	DET
m-782	60	22	normal	normal	ADJ
m-782	60	23	subgroups	subgroup	NOUN
m-782	60	24	n	n	CCONJ
m-782	60	25	of	of	ADP
m-782	60	26	g.	g.	PROPN
m-782	60	27	formally	formally	ADV
m-782	60	28	,	,	PUNCT
m-782	60	29	it	it	PRON
m-782	60	30	is	be	AUX
m-782	60	31	given	give	VERB
m-782	60	32	by	by	ADP
m-782	60	33	:	:	PUNCT
m-782	60	34	�	�	PROPN
m-782	60	35	�	�	PROPN
m-782	60	36	nil	nil	NOUN
m-782	60	37	=	=	SYM
m-782	60	38	lim←	lim←	PROPN
m-782	60	39	�	�	PROPN
m-782	60	40	/	/	SYM
m-782	60	41	�	�	PROPN
m-782	60	42	where	where	SCONJ
m-782	60	43	the	the	DET
m-782	60	44	projective	projective	ADJ
m-782	60	45	limit	limit	NOUN
m-782	60	46	is	be	AUX
m-782	60	47	taken	take	VERB
m-782	60	48	over	over	ADP
m-782	60	49	all	all	DET
m-782	60	50	normal	normal	ADJ
m-782	60	51	subgroups	subgroup	NOUN
m-782	60	52	n	n	CCONJ
m-782	60	53	of	of	ADP
m-782	60	54	g	g	NOUN
m-782	60	55	,	,	PUNCT
m-782	60	56	and	and	CCONJ
m-782	60	57	each	each	DET
m-782	60	58	g	g	NOUN
m-782	60	59	/	/	SYM
m-782	60	60	n	n	PROPN
m-782	60	61	is	be	AUX
m-782	60	62	a	a	DET
m-782	60	63	nilpotent	nilpotent	ADJ
m-782	60	64	group	group	NOUN
m-782	60	65	.	.	PUNCT
m-782	61	1	doi	doi	PROPN
m-782	61	2	10.5281	10.5281	NUM
m-782	61	3	/	/	SYM
m-782	61	4	zenodo.10511744	zenodo.10511744	PROPN
m-782	61	5	ijo	ijo	PROPN
m-782	61	6	journals	journal	NOUN
m-782	61	7	volume	volume	NOUN
m-782	61	8	07	07	NUM
m-782	62	1	|	|	NOUN
m-782	62	2	issue	issue	NOUN
m-782	62	3	01	01	NUM
m-782	63	1	|	|	CCONJ
m-782	63	2	january	january	PROPN
m-782	63	3	2024	2024	NUM
m-782	63	4	|	|	ADV
m-782	63	5	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-782	63	6	5	5	NUM
m-782	63	7	ijo	ijo	PROPN
m-782	63	8	international	international	PROPN
m-782	63	9	journal	journal	PROPN
m-782	63	10	of	of	ADP
m-782	63	11	mathematics	mathematics	PROPN
m-782	63	12	(	(	PUNCT
m-782	63	13	issn	issn	PROPN
m-782	63	14	:	:	PUNCT
m-782	63	15	2992	2992	NUM
m-782	63	16	-	-	SYM
m-782	63	17	4421	4421	NUM
m-782	63	18	)	)	PUNCT
m-782	64	1	michael	michael	PROPN
m-782	64	2	n.	n.	PROPN
m-782	64	3	john	john	PROPN
m-782	64	4	*	*	PROPN
m-782	64	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-782	64	6	volume	volume	NOUN
m-782	64	7	07	07	NUM
m-782	64	8	issue	issue	NOUN
m-782	64	9	01	01	NUM
m-782	64	10	||	||	NUM
m-782	64	11	january	january	PROPN
m-782	64	12	.	.	PUNCT
m-782	64	13	,	,	PUNCT
m-782	64	14	2024	2024	NUM
m-782	64	15	||	||	NOUN
m-782	64	16	example	example	NOUN
m-782	64	17	(	(	PUNCT
m-782	64	18	pro	pro	ADJ
m-782	64	19	-	-	ADJ
m-782	64	20	nilpotent	nilpotent	ADJ
m-782	64	21	completions	completion	NOUN
m-782	64	22	)	)	PUNCT
m-782	64	23	2.8	2.8	NUM
m-782	64	24	.	.	PUNCT
m-782	65	1	consider	consider	VERB
m-782	65	2	the	the	DET
m-782	65	3	additive	additive	ADJ
m-782	65	4	group	group	NOUN
m-782	65	5	of	of	ADP
m-782	65	6	integers	integer	NOUN
m-782	65	7	z.	z.	PROPN
m-782	66	1	the	the	DET
m-782	66	2	pro	pro	ADJ
m-782	66	3	-	-	ADJ
m-782	66	4	nilpotent	nilpotent	ADJ
m-782	66	5	completion	completion	NOUN
m-782	66	6	�	�	PROPN
m-782	66	7	�	�	PROPN
m-782	66	8	nil	nil	NOUN
m-782	66	9	is	be	AUX
m-782	66	10	obtained	obtain	VERB
m-782	66	11	by	by	ADP
m-782	66	12	considering	consider	VERB
m-782	66	13	all	all	DET
m-782	66	14	normal	normal	ADJ
m-782	66	15	subgroups	subgroup	NOUN
m-782	66	16	n	n	CCONJ
m-782	66	17	of	of	ADP
m-782	66	18	z	z	NOUN
m-782	66	19	such	such	ADJ
m-782	66	20	that	that	SCONJ
m-782	66	21	the	the	DET
m-782	66	22	quotient	quotient	NOUN
m-782	66	23	z	z	NOUN
m-782	66	24	/	/	SYM
m-782	66	25	n	n	PROPN
m-782	66	26	is	be	AUX
m-782	66	27	a	a	DET
m-782	66	28	nilpotent	nilpotent	ADJ
m-782	66	29	group	group	NOUN
m-782	66	30	.	.	PUNCT
m-782	67	1	since	since	SCONJ
m-782	67	2	every	every	DET
m-782	67	3	quotient	quotient	NOUN
m-782	67	4	z	z	NOUN
m-782	67	5	/	/	SYM
m-782	67	6	nz	nz	PROPN
m-782	67	7	is	be	AUX
m-782	67	8	nilpotent	nilpotent	ADJ
m-782	67	9	(	(	PUNCT
m-782	67	10	as	as	SCONJ
m-782	67	11	it	it	PRON
m-782	67	12	is	be	AUX
m-782	67	13	a	a	DET
m-782	67	14	cyclic	cyclic	ADJ
m-782	67	15	group	group	NOUN
m-782	67	16	of	of	ADP
m-782	67	17	prime	prime	ADJ
m-782	67	18	order	order	NOUN
m-782	67	19	)	)	PUNCT
m-782	67	20	,	,	PUNCT
m-782	67	21	the	the	DET
m-782	67	22	pro	pro	ADJ
m-782	67	23	-	-	ADJ
m-782	67	24	nilpotent	nilpotent	ADJ
m-782	67	25	completion	completion	NOUN
m-782	67	26	�	�	PROPN
m-782	67	27	�	�	PROPN
m-782	67	28	nil	nil	NOUN
m-782	67	29	is	be	AUX
m-782	67	30	the	the	DET
m-782	67	31	projective	projective	ADJ
m-782	67	32	limit	limit	NOUN
m-782	67	33	of	of	ADP
m-782	67	34	all	all	DET
m-782	67	35	these	these	DET
m-782	67	36	nilpotent	nilpotent	ADJ
m-782	67	37	quotients	quotient	NOUN
m-782	67	38	:	:	PUNCT
m-782	67	39	�	�	PROPN
m-782	67	40	�	�	PROPN
m-782	67	41	nil=	nil=	NUM
m-782	67	42	lim←	lim←	PROPN
m-782	67	43	�	�	PROPN
m-782	67	44	/	/	SYM
m-782	67	45	�	�	PROPN
m-782	67	46	�	�	PROPN
m-782	67	47	this	this	DET
m-782	67	48	pro	pro	ADJ
m-782	67	49	-	-	ADJ
m-782	67	50	nilpotent	nilpotent	ADJ
m-782	67	51	completion	completion	NOUN
m-782	67	52	can	can	AUX
m-782	67	53	be	be	AUX
m-782	67	54	identified	identify	VERB
m-782	67	55	with	with	ADP
m-782	67	56	the	the	DET
m-782	67	57	ring	ring	NOUN
m-782	67	58	of	of	ADP
m-782	67	59	p	p	NOUN
m-782	67	60	-	-	PUNCT
m-782	67	61	adic	adic	ADJ
m-782	67	62	integers	integer	NOUN
m-782	67	63	zp	zp	NOUN
m-782	67	64	,	,	PUNCT
m-782	67	65	where	where	SCONJ
m-782	67	66	p	p	NOUN
m-782	67	67	is	be	AUX
m-782	67	68	any	any	DET
m-782	67	69	prime	prime	ADJ
m-782	67	70	number	number	NOUN
m-782	67	71	.	.	PUNCT
m-782	68	1	the	the	DET
m-782	68	2	pro	pro	ADJ
m-782	68	3	-	-	ADJ
m-782	68	4	nilpotent	nilpotent	ADJ
m-782	68	5	completion	completion	NOUN
m-782	68	6	captures	capture	VERB
m-782	68	7	the	the	DET
m-782	68	8	p	p	ADJ
m-782	68	9	-	-	PUNCT
m-782	68	10	adic	adic	ADJ
m-782	68	11	topology	topology	NOUN
m-782	68	12	of	of	ADP
m-782	68	13	the	the	DET
m-782	68	14	integers	integer	NOUN
m-782	68	15	.	.	PUNCT
m-782	69	1	3.central	3.central	NUM
m-782	69	2	idea	idea	NOUN
m-782	69	3	lemma	lemma	PROPN
m-782	69	4	3.1	3.1	NUM
m-782	69	5	.	.	PUNCT
m-782	69	6	characterization	characterization	NOUN
m-782	69	7	of	of	ADP
m-782	69	8	para-	para-	NOUN
m-782	69	9	�	�	PROPN
m-782	69	10	relations	relation	NOUN
m-782	69	11	in	in	ADP
m-782	69	12	finitely	finitely	ADV
m-782	69	13	generated	generate	VERB
m-782	69	14	residually	residually	ADV
m-782	69	15	nilpotent	nilpotent	ADJ
m-782	69	16	groups	group	NOUN
m-782	69	17	.	.	PUNCT
m-782	70	1	statement	statement	NOUN
m-782	70	2	:	:	PUNCT
m-782	70	3	let	let	VERB
m-782	70	4	g	g	PRON
m-782	70	5	be	be	AUX
m-782	70	6	a	a	DET
m-782	70	7	finitely	finitely	ADV
m-782	70	8	generated	generate	VERB
m-782	70	9	residually	residually	ADV
m-782	70	10	nilpotent	nilpotent	ADJ
m-782	70	11	group	group	NOUN
m-782	70	12	.	.	PUNCT
m-782	71	1	a	a	DET
m-782	71	2	group	group	NOUN
m-782	71	3	homomorphism	homomorphism	NOUN
m-782	71	4	φ	φ	NOUN
m-782	71	5	:	:	PUNCT
m-782	71	6	g→h	g→h	PRON
m-782	71	7	is	be	AUX
m-782	71	8	a	a	DET
m-782	71	9	para-	para-	NUM
m-782	71	10	�	�	NOUN
m-782	71	11	relation	relation	NOUN
m-782	71	12	if	if	SCONJ
m-782	71	13	and	and	CCONJ
m-782	71	14	only	only	ADV
m-782	71	15	if	if	SCONJ
m-782	71	16	,	,	PUNCT
m-782	71	17	for	for	ADP
m-782	71	18	every	every	DET
m-782	71	19	finitely	finitely	ADV
m-782	71	20	generated	generate	VERB
m-782	71	21	subgroup	subgroup	NOUN
m-782	71	22	k	k	PROPN
m-782	71	23	of	of	ADP
m-782	71	24	g	g	PROPN
m-782	71	25	,	,	PUNCT
m-782	71	26	the	the	DET
m-782	71	27	kernel	kernel	NOUN
m-782	71	28	ker(φ	ker(φ	X
m-782	71	29	↾	↾	X
m-782	71	30	k	k	X
m-782	71	31	)	)	PUNCT
m-782	71	32	is	be	AUX
m-782	71	33	nilpotent	nilpotent	ADJ
m-782	71	34	.	.	PUNCT
m-782	72	1	proof	proof	NOUN
m-782	72	2	:	:	PUNCT
m-782	72	3	forward	forward	ADJ
m-782	72	4	direction	direction	NOUN
m-782	72	5	:	:	PUNCT
m-782	72	6	assume	assume	VERB
m-782	72	7	φ	φ	NOUN
m-782	72	8	:	:	PUNCT
m-782	72	9	g→h	g→h	PRON
m-782	72	10	is	be	AUX
m-782	72	11	a	a	DET
m-782	72	12	para-	para-	NUM
m-782	72	13	�	�	NOUN
m-782	72	14	relation	relation	NOUN
m-782	72	15	.	.	PUNCT
m-782	73	1	this	this	PRON
m-782	73	2	implies	imply	VERB
m-782	73	3	that	that	SCONJ
m-782	73	4	for	for	ADP
m-782	73	5	every	every	DET
m-782	73	6	normal	normal	ADJ
m-782	73	7	subgroup	subgroup	NOUN
m-782	73	8	n	n	PROPN
m-782	73	9	of	of	ADP
m-782	73	10	g	g	PROPN
m-782	73	11	,	,	PUNCT
m-782	73	12	the	the	DET
m-782	73	13	induced	induce	VERB
m-782	73	14	homomorphism	homomorphism	NOUN
m-782	73	15	φn	φn	ADP
m-782	73	16	:	:	PUNCT
m-782	73	17	g	g	PROPN
m-782	73	18	/	/	SYM
m-782	73	19	n→h	n→h	NOUN
m-782	73	20	/	/	SYM
m-782	73	21	φ(n	φ(n	NOUN
m-782	73	22	)	)	PUNCT
m-782	73	23	is	be	AUX
m-782	73	24	an	an	DET
m-782	73	25	isomorphism	isomorphism	NOUN
m-782	73	26	.	.	PUNCT
m-782	74	1	consider	consider	VERB
m-782	74	2	a	a	DET
m-782	74	3	finitely	finitely	ADV
m-782	74	4	generated	generate	VERB
m-782	74	5	subgroup	subgroup	NOUN
m-782	74	6	k	k	PROPN
m-782	74	7	of	of	ADP
m-782	74	8	g	g	PROPN
m-782	74	9	,	,	PUNCT
m-782	74	10	and	and	CCONJ
m-782	74	11	let	let	VERB
m-782	74	12	l	l	NOUN
m-782	74	13	be	be	AUX
m-782	74	14	a	a	DET
m-782	74	15	normal	normal	ADJ
m-782	74	16	subgroup	subgroup	NOUN
m-782	74	17	of	of	ADP
m-782	74	18	k.	k.	PROPN
m-782	74	19	since	since	SCONJ
m-782	74	20	k	k	PROPN
m-782	74	21	is	be	AUX
m-782	74	22	finitely	finitely	ADV
m-782	74	23	generated	generate	VERB
m-782	74	24	,	,	PUNCT
m-782	74	25	l	l	PROPN
m-782	74	26	is	be	AUX
m-782	74	27	also	also	ADV
m-782	74	28	finitely	finitely	ADV
m-782	74	29	generated	generate	VERB
m-782	74	30	.	.	PUNCT
m-782	75	1	now	now	ADV
m-782	75	2	,	,	PUNCT
m-782	75	3	consider	consider	VERB
m-782	75	4	the	the	DET
m-782	75	5	homomorphism	homomorphism	NOUN
m-782	75	6	φ	φ	X
m-782	75	7	↾	↾	NOUN
m-782	75	8	k	k	NOUN
m-782	75	9	:	:	PUNCT
m-782	75	10	k→h	k→h	NOUN
m-782	75	11	obtained	obtain	VERB
m-782	75	12	by	by	ADP
m-782	75	13	restricting	restrict	VERB
m-782	75	14	φ	φ	PROPN
m-782	75	15	to	to	ADP
m-782	75	16	k.	k.	PROPN
m-782	75	17	the	the	DET
m-782	75	18	kernel	kernel	PROPN
m-782	75	19	of	of	ADP
m-782	75	20	φ	φ	PROPN
m-782	75	21	↾	↾	PROPN
m-782	75	22	k	k	X
m-782	75	23	isker(φ	isker(φ	X
m-782	75	24	↾	↾	NOUN
m-782	75	25	k	k	X
m-782	75	26	)	)	PUNCT
m-782	75	27	=	=	PUNCT
m-782	75	28	k∩ker(φ	k∩ker(φ	PROPN
m-782	75	29	)	)	PUNCT
m-782	75	30	,	,	PUNCT
m-782	75	31	where	where	SCONJ
m-782	75	32	ker(φ	ker(φ	X
m-782	75	33	)	)	PUNCT
m-782	75	34	is	be	AUX
m-782	75	35	the	the	DET
m-782	75	36	kernel	kernel	NOUN
m-782	75	37	of	of	ADP
m-782	75	38	φ	φ	PROPN
m-782	75	39	in	in	ADP
m-782	75	40	g.	g.	PROPN
m-782	75	41	doi	doi	PROPN
m-782	75	42	10.5281	10.5281	NUM
m-782	75	43	/	/	SYM
m-782	75	44	zenodo.10511744	zenodo.10511744	PROPN
m-782	75	45	ijo	ijo	PROPN
m-782	75	46	journals	journal	NOUN
m-782	75	47	volume	volume	NOUN
m-782	75	48	07	07	NUM
m-782	76	1	|	|	NOUN
m-782	76	2	issue	issue	NOUN
m-782	76	3	01	01	NUM
m-782	77	1	|	|	CCONJ
m-782	77	2	january	january	PROPN
m-782	77	3	2024	2024	NUM
m-782	77	4	|	|	ADV
m-782	77	5	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-782	77	6	6	6	NUM
m-782	77	7	ijo	ijo	PROPN
m-782	77	8	international	international	PROPN
m-782	77	9	journal	journal	PROPN
m-782	77	10	of	of	ADP
m-782	77	11	mathematics	mathematics	PROPN
m-782	77	12	(	(	PUNCT
m-782	77	13	issn	issn	PROPN
m-782	77	14	:	:	PUNCT
m-782	77	15	2992	2992	NUM
m-782	77	16	-	-	SYM
m-782	77	17	4421	4421	NUM
m-782	77	18	)	)	PUNCT
m-782	78	1	michael	michael	PROPN
m-782	78	2	n.	n.	PROPN
m-782	78	3	john	john	PROPN
m-782	78	4	*	*	PROPN
m-782	78	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-782	78	6	volume	volume	NOUN
m-782	78	7	07	07	NUM
m-782	78	8	issue	issue	NOUN
m-782	78	9	01	01	NUM
m-782	78	10	||	||	NUM
m-782	78	11	january	january	PROPN
m-782	78	12	.	.	PUNCT
m-782	78	13	,	,	PUNCT
m-782	78	14	2024	2024	NUM
m-782	78	15	||	||	NOUN
m-782	78	16	since	since	SCONJ
m-782	78	17	φ	φ	PROPN
m-782	78	18	is	be	AUX
m-782	78	19	a	a	DET
m-782	78	20	para-	para-	NUM
m-782	78	21	�	�	NOUN
m-782	78	22	relation	relation	NOUN
m-782	78	23	,	,	PUNCT
m-782	78	24	ker(φ	ker(φ	X
m-782	78	25	)	)	PUNCT
m-782	78	26	is	be	AUX
m-782	78	27	nilpotent	nilpotent	ADJ
m-782	78	28	.	.	PUNCT
m-782	79	1	as	as	SCONJ
m-782	79	2	l	l	NOUN
m-782	79	3	is	be	AUX
m-782	79	4	a	a	DET
m-782	79	5	normal	normal	ADJ
m-782	79	6	subgroup	subgroup	NOUN
m-782	79	7	of	of	ADP
m-782	79	8	k	k	PROPN
m-782	79	9	,	,	PUNCT
m-782	79	10	l	l	PROPN
m-782	79	11	is	be	AUX
m-782	79	12	also	also	ADV
m-782	79	13	a	a	DET
m-782	79	14	normal	normal	ADJ
m-782	79	15	subgroup	subgroup	NOUN
m-782	79	16	of	of	ADP
m-782	79	17	ker(φ	ker(φ	PROPN
m-782	79	18	)	)	PUNCT
m-782	79	19	.	.	PUNCT
m-782	80	1	thus	thus	ADV
m-782	80	2	,	,	PUNCT
m-782	80	3	the	the	DET
m-782	80	4	quotient	quotient	NOUN
m-782	80	5	ker(φ)/l	ker(φ)/l	PROPN
m-782	80	6	is	be	AUX
m-782	80	7	nilpotent	nilpotent	ADJ
m-782	80	8	.	.	PUNCT
m-782	81	1	by	by	ADP
m-782	81	2	the	the	DET
m-782	81	3	correspondence	correspondence	NOUN
m-782	81	4	theorem	theorem	VERB
m-782	81	5	,	,	PUNCT
m-782	81	6	this	this	PRON
m-782	81	7	implies	imply	VERB
m-782	81	8	that	that	SCONJ
m-782	81	9	(	(	PUNCT
m-782	81	10	ker(φ)/l)∩k	ker(φ)/l)∩k	PROPN
m-782	81	11	/	/	SYM
m-782	81	12	l	l	NOUN
m-782	81	13	is	be	AUX
m-782	81	14	nilpotent	nilpotent	ADJ
m-782	81	15	.	.	PUNCT
m-782	82	1	now	now	ADV
m-782	82	2	,	,	PUNCT
m-782	82	3	consider	consider	VERB
m-782	82	4	the	the	DET
m-782	82	5	homomorphism	homomorphism	NOUN
m-782	82	6	φk	φk	ADP
m-782	82	7	/	/	SYM
m-782	82	8	l	l	NOUN
m-782	82	9	:	:	PUNCT
m-782	82	10	k	k	X
m-782	82	11	/	/	SYM
m-782	82	12	l→h	l→h	PROPN
m-782	82	13	/	/	SYM
m-782	82	14	φ(l	φ(l	PROPN
m-782	82	15	)	)	PUNCT
m-782	82	16	induced	induce	VERB
m-782	82	17	by	by	ADP
m-782	82	18	φ	φ	PROPN
m-782	82	19	on	on	ADP
m-782	82	20	the	the	DET
m-782	82	21	quotient	quotient	NOUN
m-782	82	22	group	group	NOUN
m-782	82	23	k	k	PROPN
m-782	82	24	/	/	SYM
m-782	82	25	l.	l.	PROPN
m-782	82	26	the	the	DET
m-782	82	27	kernel	kernel	NOUN
m-782	82	28	of	of	ADP
m-782	82	29	φk	φk	ADP
m-782	82	30	/	/	SYM
m-782	82	31	l	l	NOUN
m-782	82	32	is	be	AUX
m-782	82	33	(	(	PUNCT
m-782	82	34	ker(φ)/l)∩k	ker(φ)/l)∩k	PROPN
m-782	82	35	/	/	SYM
m-782	82	36	l.	l.	PROPN
m-782	82	37	since	since	SCONJ
m-782	82	38	this	this	DET
m-782	82	39	intersection	intersection	NOUN
m-782	82	40	is	be	AUX
m-782	82	41	nilpotent	nilpotent	ADJ
m-782	82	42	,	,	PUNCT
m-782	82	43	it	it	PRON
m-782	82	44	follows	follow	VERB
m-782	82	45	that	that	SCONJ
m-782	82	46	φk	φk	ADP
m-782	82	47	/	/	SYM
m-782	82	48	l	l	NOUN
m-782	82	49	is	be	AUX
m-782	82	50	an	an	DET
m-782	82	51	isomorphism	isomorphism	NOUN
m-782	82	52	.	.	PUNCT
m-782	83	1	therefore	therefore	ADV
m-782	83	2	,	,	PUNCT
m-782	83	3	φ	φ	X
m-782	83	4	↾	↾	X
m-782	83	5	k	k	X
m-782	83	6	has	have	VERB
m-782	83	7	a	a	DET
m-782	83	8	nilpotent	nilpotent	ADJ
m-782	83	9	kernel	kernel	NOUN
m-782	83	10	.	.	PUNCT
m-782	84	1	backward	backward	ADJ
m-782	84	2	direction	direction	NOUN
m-782	84	3	:	:	PUNCT
m-782	84	4	conversely	conversely	ADV
m-782	84	5	,	,	PUNCT
m-782	84	6	assume	assume	VERB
m-782	84	7	that	that	SCONJ
m-782	84	8	for	for	ADP
m-782	84	9	every	every	DET
m-782	84	10	finitely	finitely	ADV
m-782	84	11	generated	generate	VERB
m-782	84	12	subgroup	subgroup	NOUN
m-782	84	13	k	k	PROPN
m-782	84	14	of	of	ADP
m-782	84	15	g	g	PROPN
m-782	84	16	,	,	PUNCT
m-782	84	17	the	the	DET
m-782	84	18	kernel	kernel	NOUN
m-782	84	19	ker(φ	ker(φ	X
m-782	84	20	↾	↾	X
m-782	84	21	k	k	X
m-782	84	22	)	)	PUNCT
m-782	84	23	is	be	AUX
m-782	84	24	nilpotent	nilpotent	ADJ
m-782	84	25	.	.	PUNCT
m-782	85	1	we	we	PRON
m-782	85	2	need	need	VERB
m-782	85	3	to	to	PART
m-782	85	4	show	show	VERB
m-782	85	5	that	that	SCONJ
m-782	85	6	φ	φ	PROPN
m-782	85	7	is	be	AUX
m-782	85	8	a	a	DET
m-782	85	9	para-	para-	NUM
m-782	85	10	�	�	NOUN
m-782	85	11	relation	relation	NOUN
m-782	85	12	.	.	PUNCT
m-782	86	1	let	let	VERB
m-782	86	2	n	n	PRON
m-782	86	3	be	be	AUX
m-782	86	4	a	a	DET
m-782	86	5	normal	normal	ADJ
m-782	86	6	subgroup	subgroup	NOUN
m-782	86	7	of	of	ADP
m-782	86	8	g	g	PROPN
m-782	86	9	,	,	PUNCT
m-782	86	10	and	and	CCONJ
m-782	86	11	consider	consider	VERB
m-782	86	12	the	the	DET
m-782	86	13	induced	induced	ADJ
m-782	86	14	homomorphism	homomorphism	NOUN
m-782	86	15	φn	φn	ADP
m-782	86	16	:	:	PUNCT
m-782	86	17	g	g	PROPN
m-782	86	18	/	/	SYM
m-782	86	19	n→h	n→h	NOUN
m-782	86	20	/	/	SYM
m-782	86	21	φ(n	φ(n	ADJ
m-782	86	22	)	)	PUNCT
m-782	86	23	.	.	PUNCT
m-782	87	1	we	we	PRON
m-782	87	2	aim	aim	VERB
m-782	87	3	to	to	PART
m-782	87	4	show	show	VERB
m-782	87	5	that	that	SCONJ
m-782	87	6	φn	φn	NOUN
m-782	87	7	is	be	AUX
m-782	87	8	an	an	DET
m-782	87	9	isomorphism	isomorphism	NOUN
m-782	87	10	.	.	PUNCT
m-782	88	1	take	take	VERB
m-782	88	2	any	any	DET
m-782	88	3	finitely	finitely	ADV
m-782	88	4	generated	generate	VERB
m-782	88	5	subgroup	subgroup	NOUN
m-782	88	6	k	k	PROPN
m-782	88	7	/	/	SYM
m-782	88	8	n	n	PROPN
m-782	88	9	of	of	ADP
m-782	88	10	g	g	NOUN
m-782	88	11	/	/	SYM
m-782	88	12	n.	n.	NOUN
m-782	88	13	by	by	ADP
m-782	88	14	the	the	DET
m-782	88	15	correspondence	correspondence	NOUN
m-782	88	16	theorem	theorem	VERB
m-782	88	17	,	,	PUNCT
m-782	88	18	this	this	PRON
m-782	88	19	corresponds	correspond	VERB
m-782	88	20	to	to	ADP
m-782	88	21	a	a	DET
m-782	88	22	finitely	finitely	ADV
m-782	88	23	generated	generate	VERB
m-782	88	24	subgroup	subgroup	NOUN
m-782	88	25	k	k	PROPN
m-782	88	26	of	of	ADP
m-782	88	27	g	g	PROPN
m-782	88	28	containing	contain	VERB
m-782	88	29	n.	n.	NOUN
m-782	88	30	now	now	ADV
m-782	88	31	,	,	PUNCT
m-782	88	32	consider	consider	VERB
m-782	88	33	the	the	DET
m-782	88	34	homomorphism	homomorphism	NOUN
m-782	88	35	φk	φk	ADP
m-782	88	36	:	:	PUNCT
m-782	88	37	k→h	k→h	NOUN
m-782	88	38	obtained	obtain	VERB
m-782	88	39	by	by	ADP
m-782	88	40	restricting	restrict	VERB
m-782	88	41	φ	φ	PROPN
m-782	88	42	to	to	ADP
m-782	88	43	k.	k.	PROPN
m-782	88	44	by	by	ADP
m-782	88	45	assumption	assumption	NOUN
m-782	88	46	,	,	PUNCT
m-782	88	47	the	the	DET
m-782	88	48	kernel	kernel	NOUN
m-782	88	49	ker(φk	ker(φk	PROPN
m-782	88	50	)	)	PUNCT
m-782	88	51	=	=	PUNCT
m-782	88	52	k∩ker(φ	k∩ker(φ	X
m-782	88	53	)	)	PUNCT
m-782	88	54	is	be	AUX
m-782	88	55	nilpotent	nilpotent	ADJ
m-782	88	56	.	.	PUNCT
m-782	89	1	let	let	VERB
m-782	89	2	l	l	NOUN
m-782	89	3	be	be	AUX
m-782	89	4	the	the	DET
m-782	89	5	normal	normal	ADJ
m-782	89	6	subgroup	subgroup	NOUN
m-782	89	7	l	l	PROPN
m-782	89	8	=	=	PUNCT
m-782	89	9	k∩n	k∩n	PROPN
m-782	89	10	.	.	PUNCT
m-782	90	1	since	since	SCONJ
m-782	90	2	ker(φk	ker(φk	NOUN
m-782	90	3	)	)	PUNCT
m-782	90	4	is	be	AUX
m-782	90	5	nilpotent	nilpotent	ADJ
m-782	90	6	,	,	PUNCT
m-782	90	7	it	it	PRON
m-782	90	8	follows	follow	VERB
m-782	90	9	that	that	SCONJ
m-782	90	10	(	(	PUNCT
m-782	90	11	ker(φk)/l)∩(k	ker(φk)/l)∩(k	PROPN
m-782	90	12	/	/	SYM
m-782	90	13	l	l	NOUN
m-782	90	14	)	)	PUNCT
m-782	90	15	is	be	AUX
m-782	90	16	nilpotent	nilpotent	ADJ
m-782	90	17	.	.	PUNCT
m-782	91	1	now	now	ADV
m-782	91	2	,	,	PUNCT
m-782	91	3	consider	consider	VERB
m-782	91	4	the	the	DET
m-782	91	5	homomorphism	homomorphism	NOUN
m-782	91	6	φk	φk	ADP
m-782	91	7	/	/	SYM
m-782	91	8	l	l	NOUN
m-782	91	9	:	:	PUNCT
m-782	91	10	k	k	X
m-782	91	11	/	/	SYM
m-782	91	12	l→h	l→h	PROPN
m-782	91	13	/	/	SYM
m-782	91	14	φ(l	φ(l	PROPN
m-782	91	15	)	)	PUNCT
m-782	91	16	induced	induce	VERB
m-782	91	17	by	by	ADP
m-782	91	18	φ	φ	PROPN
m-782	91	19	on	on	ADP
m-782	91	20	the	the	DET
m-782	91	21	quotient	quotient	NOUN
m-782	91	22	group	group	NOUN
m-782	91	23	k	k	PROPN
m-782	91	24	/	/	SYM
m-782	91	25	l.	l.	PROPN
m-782	91	26	the	the	DET
m-782	91	27	kernel	kernel	NOUN
m-782	91	28	of	of	ADP
m-782	91	29	φk	φk	ADP
m-782	91	30	/	/	SYM
m-782	91	31	l	l	NOUN
m-782	91	32	is	be	AUX
m-782	91	33	(	(	PUNCT
m-782	91	34	ker(φk)/l)∩(k	ker(φk)/l)∩(k	PROPN
m-782	91	35	/	/	SYM
m-782	91	36	l	l	NOUN
m-782	91	37	)	)	PUNCT
m-782	91	38	,	,	PUNCT
m-782	91	39	which	which	PRON
m-782	91	40	is	be	AUX
m-782	91	41	nilpotent	nilpotent	ADJ
m-782	91	42	.	.	PUNCT
m-782	92	1	therefore	therefore	ADV
m-782	92	2	,	,	PUNCT
m-782	92	3	φk	φk	ADP
m-782	92	4	/	/	SYM
m-782	92	5	l	l	NOUN
m-782	92	6	is	be	AUX
m-782	92	7	an	an	DET
m-782	92	8	isomorphism	isomorphism	NOUN
m-782	92	9	.	.	PUNCT
m-782	93	1	since	since	SCONJ
m-782	93	2	k	k	PROPN
m-782	93	3	/	/	SYM
m-782	93	4	n	n	PRON
m-782	93	5	was	be	AUX
m-782	93	6	an	an	DET
m-782	93	7	arbitrary	arbitrary	ADJ
m-782	93	8	finitely	finitely	ADV
m-782	93	9	generated	generate	VERB
m-782	93	10	subgroup	subgroup	NOUN
m-782	93	11	of	of	ADP
m-782	93	12	g	g	PROPN
m-782	93	13	/	/	SYM
m-782	93	14	n	n	CCONJ
m-782	93	15	,	,	PUNCT
m-782	93	16	this	this	PRON
m-782	93	17	holds	hold	VERB
m-782	93	18	for	for	ADP
m-782	93	19	all	all	DET
m-782	93	20	finitely	finitely	ADV
m-782	93	21	generated	generate	VERB
m-782	93	22	subgroups	subgroup	NOUN
m-782	93	23	of	of	ADP
m-782	93	24	g	g	NOUN
m-782	93	25	/	/	SYM
m-782	93	26	n.	n.	PROPN
m-782	93	27	thus	thus	ADV
m-782	93	28	,	,	PUNCT
m-782	93	29	φn	φn	PRON
m-782	93	30	is	be	AUX
m-782	93	31	an	an	DET
m-782	93	32	isomorphism	isomorphism	NOUN
m-782	93	33	.	.	PUNCT
m-782	94	1	doi	doi	PROPN
m-782	94	2	10.5281	10.5281	NUM
m-782	94	3	/	/	SYM
m-782	94	4	zenodo.10511744	zenodo.10511744	PROPN
m-782	94	5	ijo	ijo	PROPN
m-782	94	6	journals	journal	NOUN
m-782	94	7	volume	volume	NOUN
m-782	94	8	07	07	NUM
m-782	95	1	|	|	NOUN
m-782	95	2	issue	issue	NOUN
m-782	95	3	01	01	NUM
m-782	96	1	|	|	CCONJ
m-782	96	2	january	january	PROPN
m-782	96	3	2024	2024	NUM
m-782	96	4	|	|	ADV
m-782	96	5	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-782	96	6	7	7	NUM
m-782	96	7	ijo	ijo	PROPN
m-782	96	8	international	international	PROPN
m-782	96	9	journal	journal	PROPN
m-782	96	10	of	of	ADP
m-782	96	11	mathematics	mathematics	PROPN
m-782	96	12	(	(	PUNCT
m-782	96	13	issn	issn	PROPN
m-782	96	14	:	:	PUNCT
m-782	96	15	2992	2992	NUM
m-782	96	16	-	-	SYM
m-782	96	17	4421	4421	NUM
m-782	96	18	)	)	PUNCT
m-782	97	1	michael	michael	PROPN
m-782	97	2	n.	n.	PROPN
m-782	97	3	john	john	PROPN
m-782	97	4	*	*	PROPN
m-782	97	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-782	97	6	volume	volume	NOUN
m-782	97	7	07	07	NUM
m-782	97	8	issue	issue	NOUN
m-782	97	9	01	01	NUM
m-782	97	10	||	||	NUM
m-782	97	11	january	january	PROPN
m-782	97	12	.	.	PUNCT
m-782	97	13	,	,	PUNCT
m-782	97	14	2024	2024	NUM
m-782	97	15	||	||	NOUN
m-782	97	16	since	since	SCONJ
m-782	97	17	n	n	NUM
m-782	97	18	was	be	AUX
m-782	97	19	an	an	DET
m-782	97	20	arbitrary	arbitrary	ADJ
m-782	97	21	normal	normal	ADJ
m-782	97	22	subgroup	subgroup	NOUN
m-782	97	23	of	of	ADP
m-782	97	24	g	g	PROPN
m-782	97	25	,	,	PUNCT
m-782	97	26	this	this	PRON
m-782	97	27	establishes	establish	VERB
m-782	97	28	that	that	SCONJ
m-782	97	29	φ	φ	PROPN
m-782	97	30	is	be	AUX
m-782	97	31	a	a	DET
m-782	97	32	para	para	PROPN
m-782	97	33	�	�	PROPN
m-782	97	34	relation	relation	NOUN
m-782	97	35	.	.	PUNCT
m-782	98	1	by	by	ADP
m-782	98	2	proving	prove	VERB
m-782	98	3	both	both	DET
m-782	98	4	directions	direction	NOUN
m-782	98	5	,	,	PUNCT
m-782	98	6	we	we	PRON
m-782	98	7	conclude	conclude	VERB
m-782	98	8	that	that	SCONJ
m-782	98	9	a	a	DET
m-782	98	10	group	group	NOUN
m-782	98	11	homomorphism	homomorphism	NOUN
m-782	98	12	φ	φ	NOUN
m-782	98	13	:	:	PUNCT
m-782	98	14	g→h	g→h	PRON
m-782	98	15	is	be	AUX
m-782	98	16	a	a	DET
m-782	98	17	para-	para-	NUM
m-782	98	18	�	�	NOUN
m-782	98	19	relation	relation	NOUN
m-782	98	20	if	if	SCONJ
m-782	98	21	and	and	CCONJ
m-782	98	22	only	only	ADV
m-782	98	23	if	if	SCONJ
m-782	98	24	,	,	PUNCT
m-782	98	25	for	for	ADP
m-782	98	26	every	every	DET
m-782	98	27	finitely	finitely	ADV
m-782	98	28	generated	generate	VERB
m-782	98	29	subgroup	subgroup	NOUN
m-782	98	30	k	k	PROPN
m-782	98	31	of	of	ADP
m-782	98	32	g	g	PROPN
m-782	98	33	,	,	PUNCT
m-782	98	34	the	the	DET
m-782	98	35	kernel	kernel	NOUN
m-782	98	36	ker(φ	ker(φ	X
m-782	98	37	↾	↾	X
m-782	98	38	k	k	X
m-782	98	39	)	)	PUNCT
m-782	98	40	is	be	AUX
m-782	98	41	nilpotent	nilpotent	ADJ
m-782	98	42	.	.	PUNCT
m-782	99	1	the	the	DET
m-782	99	2	lemma	lemma	PROPN
m-782	99	3	is	be	AUX
m-782	99	4	proved	prove	VERB
m-782	99	5	.	.	PUNCT
m-782	100	1	proposition	proposition	NOUN
m-782	100	2	3.2	3.2	NUM
m-782	100	3	.	.	PUNCT
m-782	101	1	sufficient	sufficient	ADJ
m-782	101	2	conditions	condition	NOUN
m-782	101	3	on	on	ADP
m-782	101	4	monomorphisms	monomorphism	NOUN
m-782	101	5	for	for	SCONJ
m-782	101	6	�	�	PROPN
m-782	101	7	to	to	PART
m-782	101	8	be	be	AUX
m-782	101	9	para-	para-	NUM
m-782	101	10	�	�	PROPN
m-782	101	11	.	.	PUNCT
m-782	101	12	statement	statement	NOUN
m-782	101	13	:	:	PUNCT
m-782	101	14	let	let	VERB
m-782	101	15	φ	φ	NOUN
m-782	101	16	:	:	PUNCT
m-782	101	17	g→h	g→h	PRON
m-782	101	18	be	be	AUX
m-782	101	19	a	a	DET
m-782	101	20	monomorphism	monomorphism	NOUN
m-782	101	21	,	,	PUNCT
m-782	101	22	where	where	SCONJ
m-782	101	23	g	g	PROPN
m-782	101	24	is	be	AUX
m-782	101	25	a	a	DET
m-782	101	26	finitely	finitely	ADV
m-782	101	27	generated	generate	VERB
m-782	101	28	residually	residually	ADV
m-782	101	29	nilpotent	nilpotent	ADJ
m-782	101	30	group	group	NOUN
m-782	101	31	,	,	PUNCT
m-782	101	32	and	and	CCONJ
m-782	101	33	h	h	NOUN
m-782	101	34	is	be	AUX
m-782	101	35	a	a	DET
m-782	101	36	group	group	NOUN
m-782	101	37	.	.	PUNCT
m-782	102	1	if	if	SCONJ
m-782	102	2	,	,	PUNCT
m-782	102	3	for	for	ADP
m-782	102	4	every	every	DET
m-782	102	5	finitely	finitely	ADV
m-782	102	6	generated	generate	VERB
m-782	102	7	subgroup	subgroup	NOUN
m-782	102	8	k	k	PROPN
m-782	102	9	of	of	ADP
m-782	102	10	g	g	PROPN
m-782	102	11	,	,	PUNCT
m-782	102	12	the	the	DET
m-782	102	13	image	image	NOUN
m-782	102	14	φ(k	φ(k	PROPN
m-782	102	15	)	)	PUNCT
m-782	102	16	is	be	AUX
m-782	102	17	a	a	DET
m-782	102	18	para-	para-	NUM
m-782	102	19	�	�	NOUN
m-782	102	20	relation	relation	NOUN
m-782	102	21	in	in	ADP
m-782	102	22	h	h	NOUN
m-782	102	23	,	,	PUNCT
m-782	102	24	then	then	ADV
m-782	102	25	h	h	PROPN
m-782	102	26	is	be	AUX
m-782	102	27	para-	para-	VERB
m-782	102	28	�	�	PROPN
m-782	102	29	.	.	PUNCT
m-782	103	1	proof	proof	NOUN
m-782	103	2	:	:	PUNCT
m-782	103	3	assume	assume	VERB
m-782	103	4	φ	φ	NOUN
m-782	103	5	:	:	PUNCT
m-782	103	6	g→h	g→h	PRON
m-782	103	7	is	be	AUX
m-782	103	8	a	a	DET
m-782	103	9	monomorphism	monomorphism	NOUN
m-782	103	10	,	,	PUNCT
m-782	103	11	where	where	SCONJ
m-782	103	12	g	g	PROPN
m-782	103	13	is	be	AUX
m-782	103	14	finitely	finitely	ADV
m-782	103	15	generated	generate	VERB
m-782	103	16	and	and	CCONJ
m-782	103	17	residually	residually	ADV
m-782	103	18	nilpotent	nilpotent	ADJ
m-782	103	19	,	,	PUNCT
m-782	103	20	and	and	CCONJ
m-782	103	21	h	h	NOUN
m-782	103	22	is	be	AUX
m-782	103	23	a	a	DET
m-782	103	24	group	group	NOUN
m-782	103	25	.	.	PUNCT
m-782	104	1	suppose	suppose	VERB
m-782	104	2	that	that	SCONJ
m-782	104	3	for	for	ADP
m-782	104	4	every	every	DET
m-782	104	5	finitely	finitely	ADV
m-782	104	6	generated	generate	VERB
m-782	104	7	subgroup	subgroup	NOUN
m-782	104	8	k	k	PROPN
m-782	104	9	of	of	ADP
m-782	104	10	g	g	PROPN
m-782	104	11	,	,	PUNCT
m-782	104	12	the	the	DET
m-782	104	13	image	image	NOUN
m-782	104	14	φ(k	φ(k	PROPN
m-782	104	15	)	)	PUNCT
m-782	104	16	is	be	AUX
m-782	104	17	a	a	DET
m-782	104	18	para-	para-	NUM
m-782	104	19	�	�	NOUN
m-782	104	20	relation	relation	NOUN
m-782	104	21	in	in	ADP
m-782	104	22	h.	h.	PROPN
m-782	104	23	we	we	PRON
m-782	104	24	aim	aim	VERB
m-782	104	25	to	to	PART
m-782	104	26	show	show	VERB
m-782	104	27	that	that	SCONJ
m-782	104	28	h	h	NOUN
m-782	104	29	is	be	AUX
m-782	104	30	para-	para-	VERB
m-782	104	31	�	�	PROPN
m-782	104	32	.	.	PUNCT
m-782	105	1	let	let	VERB
m-782	105	2	n	n	PRON
m-782	105	3	be	be	AUX
m-782	105	4	a	a	DET
m-782	105	5	normal	normal	ADJ
m-782	105	6	subgroup	subgroup	NOUN
m-782	105	7	of	of	ADP
m-782	105	8	h	h	NOUN
m-782	105	9	,	,	PUNCT
m-782	105	10	and	and	CCONJ
m-782	105	11	consider	consider	VERB
m-782	105	12	the	the	DET
m-782	105	13	induced	induced	ADJ
m-782	105	14	homomorphism	homomorphism	NOUN
m-782	105	15	φn	φn	ADP
m-782	105	16	:	:	PUNCT
m-782	105	17	g	g	NOUN
m-782	105	18	/	/	SYM
m-782	105	19	ker(φ)→h	ker(φ)→h	NOUN
m-782	105	20	/	/	SYM
m-782	105	21	n.	n.	NOUN
m-782	105	22	we	we	PRON
m-782	105	23	need	need	VERB
m-782	105	24	to	to	PART
m-782	105	25	show	show	VERB
m-782	105	26	that	that	SCONJ
m-782	105	27	φn	φn	NOUN
m-782	105	28	is	be	AUX
m-782	105	29	an	an	DET
m-782	105	30	isomorphism	isomorphism	NOUN
m-782	105	31	.	.	PUNCT
m-782	106	1	consider	consider	VERB
m-782	106	2	any	any	DET
m-782	106	3	finitely	finitely	ADV
m-782	106	4	generated	generate	VERB
m-782	106	5	subgroup	subgroup	NOUN
m-782	106	6	k	k	PROPN
m-782	106	7	/	/	SYM
m-782	106	8	ker(φ	ker(φ	NOUN
m-782	106	9	)	)	PUNCT
m-782	106	10	of	of	ADP
m-782	106	11	g	g	NOUN
m-782	106	12	/	/	SYM
m-782	106	13	ker(φ	ker(φ	NOUN
m-782	106	14	)	)	PUNCT
m-782	106	15	.	.	PUNCT
m-782	107	1	by	by	ADP
m-782	107	2	the	the	DET
m-782	107	3	correspondence	correspondence	NOUN
m-782	107	4	theorem	theorem	VERB
m-782	107	5	,	,	PUNCT
m-782	107	6	this	this	PRON
m-782	107	7	corresponds	correspond	VERB
m-782	107	8	to	to	ADP
m-782	107	9	a	a	DET
m-782	107	10	finitely	finitely	ADV
m-782	107	11	generated	generate	VERB
m-782	107	12	subgroup	subgroup	NOUN
m-782	107	13	k	k	PROPN
m-782	107	14	of	of	ADP
m-782	107	15	g	g	PROPN
m-782	107	16	containing	contain	VERB
m-782	107	17	ker(φ	ker(φ	NOUN
m-782	107	18	)	)	PUNCT
m-782	107	19	.	.	PUNCT
m-782	108	1	now	now	ADV
m-782	108	2	,	,	PUNCT
m-782	108	3	the	the	DET
m-782	108	4	image	image	NOUN
m-782	108	5	φ(k	φ(k	PROPN
m-782	108	6	)	)	PUNCT
m-782	108	7	is	be	AUX
m-782	108	8	a	a	DET
m-782	108	9	para-	para-	NUM
m-782	108	10	�	�	NOUN
m-782	108	11	relation	relation	NOUN
m-782	108	12	in	in	ADP
m-782	108	13	h	h	NOUN
m-782	108	14	,	,	PUNCT
m-782	108	15	as	as	ADP
m-782	108	16	per	per	ADP
m-782	108	17	our	our	PRON
m-782	108	18	assumption	assumption	NOUN
m-782	108	19	.	.	PUNCT
m-782	109	1	therefore	therefore	ADV
m-782	109	2	,	,	PUNCT
m-782	109	3	the	the	DET
m-782	109	4	induced	induce	VERB
m-782	109	5	homomorphism	homomorphism	PROPN
m-782	109	6	φk	φk	ADP
m-782	109	7	:	:	PUNCT
m-782	109	8	k→h	k→h	NOUN
m-782	109	9	obtained	obtain	VERB
m-782	109	10	by	by	ADP
m-782	109	11	restricting	restrict	VERB
m-782	109	12	φ	φ	PROPN
m-782	109	13	to	to	ADP
m-782	109	14	k	k	PROPN
m-782	109	15	is	be	AUX
m-782	109	16	a	a	DET
m-782	109	17	para-	para-	NUM
m-782	109	18	�	�	NOUN
m-782	109	19	relation	relation	NOUN
m-782	109	20	in	in	ADP
m-782	109	21	h.	h.	PROPN
m-782	109	22	this	this	PRON
m-782	109	23	implies	imply	VERB
m-782	109	24	that	that	SCONJ
m-782	109	25	the	the	DET
m-782	109	26	induced	induced	ADJ
m-782	109	27	homomorphism	homomorphism	PROPN
m-782	109	28	φk	φk	ADP
m-782	109	29	/	/	SYM
m-782	109	30	ker(φ	ker(φ	PROPN
m-782	109	31	)	)	PUNCT
m-782	109	32	:	:	PUNCT
m-782	109	33	k	k	X
m-782	109	34	/	/	SYM
m-782	109	35	ker(φ)→φ(k	ker(φ)→φ(k	PROPN
m-782	109	36	)	)	PUNCT
m-782	109	37	is	be	AUX
m-782	109	38	an	an	DET
m-782	109	39	isomorphism	isomorphism	NOUN
m-782	109	40	.	.	PUNCT
m-782	110	1	doi	doi	PROPN
m-782	110	2	10.5281	10.5281	NUM
m-782	110	3	/	/	SYM
m-782	110	4	zenodo.10511744	zenodo.10511744	PROPN
m-782	110	5	ijo	ijo	PROPN
m-782	110	6	journals	journal	NOUN
m-782	110	7	volume	volume	NOUN
m-782	110	8	07	07	NUM
m-782	111	1	|	|	NOUN
m-782	111	2	issue	issue	NOUN
m-782	111	3	01	01	NUM
m-782	112	1	|	|	CCONJ
m-782	112	2	january	january	PROPN
m-782	112	3	2024	2024	NUM
m-782	112	4	|	|	ADV
m-782	112	5	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-782	112	6	8	8	NUM
m-782	112	7	ijo	ijo	PROPN
m-782	112	8	international	international	PROPN
m-782	112	9	journal	journal	PROPN
m-782	112	10	of	of	ADP
m-782	112	11	mathematics	mathematics	PROPN
m-782	112	12	(	(	PUNCT
m-782	112	13	issn	issn	PROPN
m-782	112	14	:	:	PUNCT
m-782	112	15	2992	2992	NUM
m-782	112	16	-	-	SYM
m-782	112	17	4421	4421	NUM
m-782	112	18	)	)	PUNCT
m-782	113	1	michael	michael	PROPN
m-782	113	2	n.	n.	PROPN
m-782	113	3	john	john	PROPN
m-782	113	4	*	*	PROPN
m-782	113	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-782	113	6	volume	volume	NOUN
m-782	113	7	07	07	NUM
m-782	113	8	issue	issue	NOUN
m-782	113	9	01	01	NUM
m-782	113	10	||	||	NUM
m-782	113	11	january	january	PROPN
m-782	113	12	.	.	PUNCT
m-782	113	13	,	,	PUNCT
m-782	113	14	2024	2024	NUM
m-782	113	15	||	||	VERB
m-782	113	16	now	now	ADV
m-782	113	17	,	,	PUNCT
m-782	113	18	consider	consider	VERB
m-782	113	19	the	the	DET
m-782	113	20	homomorphism	homomorphism	NOUN
m-782	113	21	φk	φk	ADP
m-782	113	22	/	/	SYM
m-782	113	23	n	n	CCONJ
m-782	113	24	:	:	PUNCT
m-782	113	25	k	k	X
m-782	113	26	/	/	SYM
m-782	113	27	n→h	n→h	PROPN
m-782	113	28	/	/	SYM
m-782	113	29	n	n	PRON
m-782	113	30	induced	induce	VERB
m-782	113	31	by	by	ADP
m-782	113	32	φ	φ	PROPN
m-782	113	33	on	on	ADP
m-782	113	34	the	the	DET
m-782	113	35	quotient	quotient	NOUN
m-782	113	36	group	group	NOUN
m-782	113	37	k	k	PROPN
m-782	113	38	/	/	SYM
m-782	113	39	n.	n.	NOUN
m-782	113	40	this	this	PRON
m-782	113	41	is	be	AUX
m-782	113	42	the	the	DET
m-782	113	43	composition	composition	NOUN
m-782	113	44	of	of	ADP
m-782	113	45	the	the	DET
m-782	113	46	isomorphism	isomorphism	NOUN
m-782	113	47	φk	φk	ADP
m-782	113	48	/	/	SYM
m-782	113	49	ker(φ	ker(φ	NOUN
m-782	113	50	)	)	PUNCT
m-782	113	51	and	and	CCONJ
m-782	113	52	the	the	DET
m-782	113	53	natural	natural	ADJ
m-782	113	54	projection	projection	NOUN
m-782	113	55	k	k	PROPN
m-782	113	56	/	/	SYM
m-782	113	57	ker(φ)→k	ker(φ)→k	NOUN
m-782	113	58	/	/	SYM
m-782	113	59	n.	n.	NOUN
m-782	113	60	since	since	SCONJ
m-782	113	61	compositions	composition	NOUN
m-782	113	62	of	of	ADP
m-782	113	63	isomorphisms	isomorphism	NOUN
m-782	113	64	are	be	AUX
m-782	113	65	isomorphisms	isomorphism	NOUN
m-782	113	66	,	,	PUNCT
m-782	113	67	φk	φk	PROPN
m-782	113	68	/	/	SYM
m-782	113	69	n	n	PRON
m-782	113	70	is	be	AUX
m-782	113	71	an	an	DET
m-782	113	72	isomorphism	isomorphism	NOUN
m-782	113	73	.	.	PUNCT
m-782	114	1	since	since	SCONJ
m-782	114	2	k	k	PROPN
m-782	114	3	/	/	SYM
m-782	114	4	n	n	PRON
m-782	114	5	was	be	AUX
m-782	114	6	an	an	DET
m-782	114	7	arbitrary	arbitrary	ADJ
m-782	114	8	finitely	finitely	ADV
m-782	114	9	generated	generate	VERB
m-782	114	10	subgroup	subgroup	NOUN
m-782	114	11	of	of	ADP
m-782	114	12	g	g	PROPN
m-782	114	13	/	/	SYM
m-782	114	14	ker(φ	ker(φ	NOUN
m-782	114	15	)	)	PUNCT
m-782	114	16	,	,	PUNCT
m-782	114	17	this	this	PRON
m-782	114	18	holds	hold	VERB
m-782	114	19	for	for	ADP
m-782	114	20	all	all	DET
m-782	114	21	finitely	finitely	ADV
m-782	114	22	generated	generate	VERB
m-782	114	23	subgroups	subgroup	NOUN
m-782	114	24	of	of	ADP
m-782	114	25	g	g	NOUN
m-782	114	26	/	/	SYM
m-782	114	27	ker(φ	ker(φ	NOUN
m-782	114	28	)	)	PUNCT
m-782	114	29	.	.	PUNCT
m-782	115	1	thus	thus	ADV
m-782	115	2	,	,	PUNCT
m-782	115	3	φn	φn	PRON
m-782	115	4	is	be	AUX
m-782	115	5	an	an	DET
m-782	115	6	isomorphism	isomorphism	NOUN
m-782	115	7	.	.	PUNCT
m-782	116	1	since	since	SCONJ
m-782	116	2	n	n	NUM
m-782	116	3	was	be	AUX
m-782	116	4	an	an	DET
m-782	116	5	arbitrary	arbitrary	ADJ
m-782	116	6	normal	normal	ADJ
m-782	116	7	subgroup	subgroup	NOUN
m-782	116	8	of	of	ADP
m-782	116	9	h	h	NOUN
m-782	116	10	,	,	PUNCT
m-782	116	11	this	this	PRON
m-782	116	12	establishes	establish	VERB
m-782	116	13	that	that	SCONJ
m-782	116	14	h	h	NOUN
m-782	116	15	is	be	AUX
m-782	116	16	para-	para-	VERB
m-782	116	17	�	�	PROPN
m-782	116	18	.	.	PUNCT
m-782	116	19	by	by	ADP
m-782	116	20	proving	prove	VERB
m-782	116	21	the	the	DET
m-782	116	22	sufficiency	sufficiency	NOUN
m-782	116	23	of	of	ADP
m-782	116	24	the	the	DET
m-782	116	25	conditions	condition	NOUN
m-782	116	26	on	on	ADP
m-782	116	27	monomorphisms	monomorphism	NOUN
m-782	116	28	for	for	SCONJ
m-782	116	29	h	h	NOUN
m-782	116	30	to	to	PART
m-782	116	31	be	be	AUX
m-782	116	32	para	para	PROPN
m-782	116	33	�	�	PROPN
m-782	116	34	,	,	PUNCT
m-782	116	35	the	the	DET
m-782	116	36	proposition	proposition	NOUN
m-782	116	37	is	be	AUX
m-782	116	38	proved	prove	VERB
m-782	116	39	.	.	PUNCT
m-782	117	1	theorem	theorem	VERB
m-782	117	2	3.3	3.3	NUM
m-782	117	3	.	.	PUNCT
m-782	118	1	implications	implication	NOUN
m-782	118	2	of	of	ADP
m-782	118	3	para-	para-	NOUN
m-782	118	4	�	�	PROPN
m-782	118	5	relations	relation	NOUN
m-782	118	6	on	on	ADP
m-782	118	7	the	the	DET
m-782	118	8	hirsch	hirsch	PROPN
m-782	118	9	length	length	NOUN
m-782	118	10	of	of	ADP
m-782	118	11	certain	certain	ADJ
m-782	118	12	polycyclic	polycyclic	NOUN
m-782	118	13	groups	group	NOUN
m-782	118	14	.	.	PUNCT
m-782	119	1	statement	statement	NOUN
m-782	119	2	:	:	PUNCT
m-782	119	3	let	let	VERB
m-782	119	4	g	g	PRON
m-782	119	5	be	be	AUX
m-782	119	6	a	a	DET
m-782	119	7	finitely	finitely	ADV
m-782	119	8	generated	generate	VERB
m-782	119	9	residually	residually	ADV
m-782	119	10	nilpotent	nilpotent	ADJ
m-782	119	11	group	group	NOUN
m-782	119	12	with	with	ADP
m-782	119	13	a	a	DET
m-782	119	14	para-	para-	NUM
m-782	119	15	�	�	NOUN
m-782	119	16	relation	relation	NOUN
m-782	119	17	in	in	ADP
m-782	119	18	its	its	PRON
m-782	119	19	subgroup	subgroup	NOUN
m-782	119	20	h.	h.	NOUN
m-782	120	1	if	if	SCONJ
m-782	120	2	g	g	PROPN
m-782	120	3	is	be	AUX
m-782	120	4	polycyclic	polycyclic	NOUN
m-782	120	5	,	,	PUNCT
m-782	120	6	then	then	ADV
m-782	120	7	the	the	DET
m-782	120	8	hirsch	hirsch	PROPN
m-782	120	9	length	length	NOUN
m-782	120	10	of	of	ADP
m-782	120	11	g	g	PROPN
m-782	120	12	is	be	AUX
m-782	120	13	bounded	bound	VERB
m-782	120	14	by	by	ADP
m-782	120	15	the	the	DET
m-782	120	16	hirsch	hirsch	PROPN
m-782	120	17	length	length	NOUN
m-782	120	18	of	of	ADP
m-782	120	19	h.	h.	PROPN
m-782	120	20	proof	proof	NOUN
m-782	120	21	:	:	PUNCT
m-782	120	22	assume	assume	VERB
m-782	120	23	g	g	PROPN
m-782	120	24	is	be	AUX
m-782	120	25	a	a	DET
m-782	120	26	finitely	finitely	ADV
m-782	120	27	generated	generate	VERB
m-782	120	28	residually	residually	ADV
m-782	120	29	nilpotent	nilpotent	ADJ
m-782	120	30	group	group	NOUN
m-782	120	31	with	with	ADP
m-782	120	32	a	a	DET
m-782	120	33	para-	para-	NUM
m-782	120	34	�	�	NOUN
m-782	120	35	relation	relation	NOUN
m-782	120	36	in	in	ADP
m-782	120	37	its	its	PRON
m-782	120	38	subgroup	subgroup	NOUN
m-782	120	39	h.	h.	PROPN
m-782	120	40	suppose	suppose	VERB
m-782	120	41	g	g	PROPN
m-782	120	42	is	be	AUX
m-782	120	43	polycyclic	polycyclic	ADJ
m-782	120	44	.	.	PUNCT
m-782	121	1	we	we	PRON
m-782	121	2	aim	aim	VERB
m-782	121	3	to	to	PART
m-782	121	4	show	show	VERB
m-782	121	5	that	that	SCONJ
m-782	121	6	the	the	DET
m-782	121	7	hirsch	hirsch	PROPN
m-782	121	8	length	length	NOUN
m-782	121	9	of	of	ADP
m-782	121	10	g	g	PROPN
m-782	121	11	is	be	AUX
m-782	121	12	bounded	bound	VERB
m-782	121	13	by	by	ADP
m-782	121	14	the	the	DET
m-782	121	15	hirsch	hirsch	PROPN
m-782	121	16	length	length	NOUN
m-782	121	17	of	of	ADP
m-782	121	18	h.	h.	PROPN
m-782	121	19	recall	recall	PROPN
m-782	121	20	that	that	SCONJ
m-782	121	21	the	the	DET
m-782	121	22	hirsch	hirsch	PROPN
m-782	121	23	length	length	NOUN
m-782	121	24	of	of	ADP
m-782	121	25	a	a	DET
m-782	121	26	group	group	NOUN
m-782	121	27	is	be	AUX
m-782	121	28	a	a	DET
m-782	121	29	measure	measure	NOUN
m-782	121	30	of	of	ADP
m-782	121	31	the	the	DET
m-782	121	32	growth	growth	NOUN
m-782	121	33	rate	rate	NOUN
m-782	121	34	of	of	ADP
m-782	121	35	its	its	PRON
m-782	121	36	lower	low	ADJ
m-782	121	37	central	central	ADJ
m-782	121	38	series	series	NOUN
m-782	121	39	.	.	PUNCT
m-782	122	1	let	let	VERB
m-782	122	2	g	g	NOUN
m-782	122	3	=	=	SYM
m-782	122	4	⟨g1	⟨g1	PROPN
m-782	122	5	,	,	PUNCT
m-782	122	6	g2	g2	PROPN
m-782	122	7	,	,	PUNCT
m-782	122	8	…	…	PUNCT
m-782	122	9	,	,	PUNCT
m-782	122	10	gn⟩	gn⟩	NOUN
m-782	122	11	be	be	AUX
m-782	122	12	a	a	DET
m-782	122	13	generating	generate	VERB
m-782	122	14	set	set	NOUN
m-782	122	15	for	for	ADP
m-782	122	16	g.	g.	PROPN
m-782	122	17	since	since	SCONJ
m-782	122	18	g	g	PROPN
m-782	122	19	is	be	AUX
m-782	122	20	polycyclic	polycyclic	NOUN
m-782	122	21	,	,	PUNCT
m-782	122	22	it	it	PRON
m-782	122	23	has	have	VERB
m-782	122	24	a	a	DET
m-782	122	25	subnormal	subnormal	ADJ
m-782	122	26	series	series	NOUN
m-782	122	27	doi	doi	PROPN
m-782	122	28	10.5281	10.5281	NUM
m-782	122	29	/	/	SYM
m-782	122	30	zenodo.10511744	zenodo.10511744	PROPN
m-782	122	31	ijo	ijo	PROPN
m-782	122	32	journals	journal	NOUN
m-782	122	33	volume	volume	NOUN
m-782	122	34	07	07	NUM
m-782	122	35	|	|	NOUN
m-782	122	36	issue	issue	NOUN
m-782	122	37	01	01	NUM
m-782	123	1	|	|	CCONJ
m-782	123	2	january	january	PROPN
m-782	123	3	2024	2024	NUM
m-782	123	4	|	|	ADV
m-782	123	5	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-782	123	6	9	9	NUM
m-782	123	7	ijo	ijo	PROPN
m-782	123	8	international	international	PROPN
m-782	123	9	journal	journal	PROPN
m-782	123	10	of	of	ADP
m-782	123	11	mathematics	mathematics	PROPN
m-782	123	12	(	(	PUNCT
m-782	123	13	issn	issn	PROPN
m-782	123	14	:	:	PUNCT
m-782	123	15	2992	2992	NUM
m-782	123	16	-	-	SYM
m-782	123	17	4421	4421	NUM
m-782	123	18	)	)	PUNCT
m-782	124	1	michael	michael	PROPN
m-782	124	2	n.	n.	PROPN
m-782	124	3	john	john	PROPN
m-782	124	4	*	*	PROPN
m-782	124	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-782	124	6	volume	volume	NOUN
m-782	124	7	07	07	NUM
m-782	124	8	issue	issue	NOUN
m-782	124	9	01	01	NUM
m-782	124	10	||	||	NUM
m-782	124	11	january	january	PROPN
m-782	124	12	.	.	PUNCT
m-782	125	1	,	,	PUNCT
m-782	125	2	2024	2024	NUM
m-782	125	3	||	||	NOUN
m-782	125	4	1	1	NUM
m-782	125	5	=	=	SYM
m-782	125	6	g0	g0	PROPN
m-782	125	7	⊴	⊴	ADP
m-782	125	8	g1	g1	PROPN
m-782	125	9	⊴	⊴	ADP
m-782	125	10	…	…	PUNCT
m-782	125	11	⊴	⊴	NUM
m-782	125	12	gk	gk	NOUN
m-782	125	13	=	=	SYM
m-782	125	14	g	g	PROPN
m-782	125	15	,	,	PUNCT
m-782	125	16	where	where	SCONJ
m-782	125	17	each	each	DET
m-782	125	18	factor	factor	NOUN
m-782	125	19	group	group	NOUN
m-782	125	20	gi+1	gi+1	PROPN
m-782	125	21	/	/	SYM
m-782	125	22	gi	gi	NOUN
m-782	125	23	is	be	AUX
m-782	125	24	cyclic	cyclic	ADJ
m-782	125	25	.	.	PUNCT
m-782	126	1	consider	consider	VERB
m-782	126	2	the	the	DET
m-782	126	3	subgroup	subgroup	NOUN
m-782	126	4	h′	h′	NOUN
m-782	126	5	=	=	PUNCT
m-782	126	6	⟨φ(g1	⟨φ(g1	PROPN
m-782	126	7	)	)	PUNCT
m-782	126	8	,	,	PUNCT
m-782	126	9	φ(g2),	φ(g2),	NOUN
m-782	126	10	…	…	PUNCT
m-782	126	11	,φ(gn)⟩	,φ(gn)⟩	PUNCT
m-782	126	12	of	of	ADP
m-782	126	13	h	h	NOUN
m-782	126	14	,	,	PUNCT
m-782	126	15	where	where	SCONJ
m-782	126	16	φ	φ	NOUN
m-782	126	17	:	:	PUNCT
m-782	126	18	g→h	g→h	PRON
m-782	126	19	is	be	AUX
m-782	126	20	the	the	DET
m-782	126	21	para-	para-	NUM
m-782	126	22	�	�	PROPN
m-782	126	23	relation	relation	NOUN
m-782	126	24	.	.	PUNCT
m-782	127	1	since	since	SCONJ
m-782	127	2	h	h	NOUN
m-782	127	3	is	be	AUX
m-782	127	4	para-	para-	VERB
m-782	127	5	�	�	PROPN
m-782	127	6	,	,	PUNCT
m-782	127	7	the	the	DET
m-782	127	8	hirsch	hirsch	PROPN
m-782	127	9	length	length	NOUN
m-782	127	10	of	of	ADP
m-782	127	11	h	h	NOUN
m-782	127	12	is	be	AUX
m-782	127	13	finite	finite	ADJ
m-782	127	14	.	.	PUNCT
m-782	128	1	now	now	ADV
m-782	128	2	,	,	PUNCT
m-782	128	3	consider	consider	VERB
m-782	128	4	the	the	DET
m-782	128	5	induced	induced	ADJ
m-782	128	6	homomorphism	homomorphism	NOUN
m-782	128	7	φi	φi	ADP
m-782	128	8	:	:	PUNCT
m-782	128	9	gi→h′	gi→h′	VERB
m-782	128	10	for	for	ADP
m-782	128	11	each	each	DET
m-782	128	12	i=0,1,	i=0,1,	NOUN
m-782	128	13	…	…	PUNCT
m-782	128	14	,k	,k	PUNCT
m-782	128	15	.	.	PUNCT
m-782	129	1	since	since	SCONJ
m-782	129	2	gi	gi	PROPN
m-782	129	3	is	be	AUX
m-782	129	4	normal	normal	ADJ
m-782	129	5	in	in	ADP
m-782	129	6	gi+1	gi+1	PROPN
m-782	129	7	,	,	PUNCT
m-782	129	8	the	the	DET
m-782	129	9	factor	factor	NOUN
m-782	129	10	group	group	NOUN
m-782	129	11	gi+1	gi+1	PROPN
m-782	129	12	/	/	SYM
m-782	129	13	gi	gi	NOUN
m-782	129	14	is	be	AUX
m-782	129	15	cyclic	cyclic	ADJ
m-782	129	16	,	,	PUNCT
m-782	129	17	and	and	CCONJ
m-782	129	18	φi(gi+1	φi(gi+1	NOUN
m-782	129	19	)	)	PUNCT
m-782	129	20	is	be	AUX
m-782	129	21	cyclic	cyclic	ADJ
m-782	129	22	in	in	ADP
m-782	129	23	h′.	h′.	PROPN
m-782	129	24	therefore	therefore	ADV
m-782	129	25	,	,	PUNCT
m-782	129	26	h′	h′	PROPN
m-782	129	27	also	also	ADV
m-782	129	28	has	have	VERB
m-782	129	29	a	a	DET
m-782	129	30	subnormal	subnormal	ADJ
m-782	129	31	series	series	NOUN
m-782	129	32	1	1	NUM
m-782	129	33	=	=	NOUN
m-782	130	1	h0′	h0′	NOUN
m-782	130	2	⊴	⊴	ADP
m-782	130	3	h1′	h1′	ADV
m-782	130	4	⊴	⊴	ADP
m-782	130	5	…	…	PUNCT
m-782	130	6	⊴	⊴	NUM
m-782	130	7	hk′	hk′	X
m-782	130	8	=	=	SYM
m-782	130	9	h′	h′	PROPN
m-782	130	10	,	,	PUNCT
m-782	130	11	where	where	SCONJ
m-782	130	12	each	each	DET
m-782	130	13	factor	factor	NOUN
m-782	130	14	group	group	NOUN
m-782	130	15	h′i+1	h′i+1	PROPN
m-782	130	16	/	/	SYM
m-782	130	17	hi′	hi′	PROPN
m-782	130	18	is	be	AUX
m-782	130	19	cyclic	cyclic	ADJ
m-782	130	20	.	.	PUNCT
m-782	131	1	since	since	SCONJ
m-782	131	2	the	the	DET
m-782	131	3	hirsch	hirsch	PROPN
m-782	131	4	length	length	NOUN
m-782	131	5	of	of	ADP
m-782	131	6	h′	h′	PROPN
m-782	131	7	is	be	AUX
m-782	131	8	finite	finite	PROPN
m-782	131	9	,	,	PUNCT
m-782	131	10	the	the	DET
m-782	131	11	subnormal	subnormal	ADJ
m-782	131	12	series	series	NOUN
m-782	131	13	of	of	ADP
m-782	131	14	h′	h′	PROPN
m-782	131	15	stabilizes	stabilize	VERB
m-782	131	16	,	,	PUNCT
m-782	131	17	i.e.	i.e.	X
m-782	131	18	,	,	PUNCT
m-782	131	19	there	there	PRON
m-782	131	20	exists	exist	VERB
m-782	131	21	i0	i0	PROPN
m-782	131	22	such	such	ADJ
m-782	131	23	that	that	SCONJ
m-782	131	24	hi′	hi′	PROPN
m-782	132	1	=	=	NOUN
m-782	132	2	h′i0	h′i0	VERB
m-782	132	3	for	for	ADP
m-782	132	4	all	all	DET
m-782	132	5	i≥i0	i≥i0	PROPN
m-782	132	6	.	.	PUNCT
m-782	133	1	correspondingly	correspondingly	ADV
m-782	133	2	,	,	PUNCT
m-782	133	3	the	the	DET
m-782	133	4	subnormal	subnormal	ADJ
m-782	133	5	series	series	NOUN
m-782	133	6	of	of	ADP
m-782	133	7	g	g	PROPN
m-782	133	8	stabilizes	stabilize	VERB
m-782	133	9	at	at	ADP
m-782	133	10	i0	i0	PROPN
m-782	133	11	,	,	PUNCT
m-782	133	12	i.e.	i.e.	X
m-782	133	13	,	,	PUNCT
m-782	133	14	gi	gi	NOUN
m-782	133	15	=	=	SYM
m-782	133	16	gi0	gi0	NOUN
m-782	133	17	for	for	ADP
m-782	133	18	all	all	DET
m-782	133	19	i≥i0	i≥i0	PROPN
m-782	133	20	.	.	PUNCT
m-782	134	1	this	this	PRON
m-782	134	2	implies	imply	VERB
m-782	134	3	that	that	SCONJ
m-782	134	4	the	the	DET
m-782	134	5	hirsch	hirsch	PROPN
m-782	134	6	length	length	NOUN
m-782	134	7	of	of	ADP
m-782	134	8	g	g	PROPN
m-782	134	9	is	be	AUX
m-782	134	10	bounded	bound	VERB
m-782	134	11	by	by	ADP
m-782	134	12	the	the	DET
m-782	134	13	hirsch	hirsch	PROPN
m-782	134	14	length	length	NOUN
m-782	134	15	of	of	ADP
m-782	134	16	h′	h′	PROPN
m-782	134	17	,	,	PUNCT
m-782	134	18	which	which	PRON
m-782	134	19	is	be	AUX
m-782	134	20	finite	finite	ADJ
m-782	134	21	.	.	PUNCT
m-782	135	1	therefore	therefore	ADV
m-782	135	2	,	,	PUNCT
m-782	135	3	the	the	DET
m-782	135	4	theorem	theorem	NOUN
m-782	135	5	is	be	AUX
m-782	135	6	proved	prove	VERB
m-782	135	7	.	.	PUNCT
m-782	136	1	theorem	theorem	VERB
m-782	136	2	3.4	3.4	NUM
m-782	136	3	.	.	PUNCT
m-782	137	1	locally	locally	ADV
m-782	137	2	polycyclic	polycyclic	ADJ
m-782	137	3	nature	nature	NOUN
m-782	137	4	of	of	ADP
m-782	137	5	pro	pro	ADJ
m-782	137	6	-	-	ADJ
m-782	137	7	nilpotent	nilpotent	ADJ
m-782	137	8	completions	completion	NOUN
m-782	137	9	of	of	ADP
m-782	137	10	specific	specific	ADJ
m-782	137	11	polycyclic	polycyclic	NOUN
m-782	137	12	groups	group	NOUN
m-782	137	13	.	.	PUNCT
m-782	138	1	statement	statement	NOUN
m-782	138	2	:	:	PUNCT
m-782	138	3	let	let	VERB
m-782	138	4	g	g	PRON
m-782	138	5	be	be	AUX
m-782	138	6	a	a	DET
m-782	138	7	polycyclic	polycyclic	NOUN
m-782	138	8	group	group	NOUN
m-782	138	9	.	.	PUNCT
m-782	139	1	the	the	DET
m-782	139	2	pro	pro	ADJ
m-782	139	3	-	-	ADJ
m-782	139	4	nilpotent	nilpotent	ADJ
m-782	139	5	completion	completion	NOUN
m-782	139	6	of	of	ADP
m-782	139	7	g	g	NOUN
m-782	139	8	with	with	ADP
m-782	139	9	respect	respect	NOUN
m-782	139	10	to	to	ADP
m-782	139	11	the	the	DET
m-782	139	12	pro	pro	ADJ
m-782	139	13	-	-	ADJ
m-782	139	14	nilpotent	nilpotent	ADJ
m-782	139	15	topology	topology	NOUN
m-782	139	16	is	be	AUX
m-782	139	17	locally	locally	ADV
m-782	139	18	polycyclic	polycyclic	NOUN
m-782	139	19	.	.	PUNCT
m-782	140	1	proof	proof	NOUN
m-782	140	2	:	:	PUNCT
m-782	140	3	consider	consider	VERB
m-782	140	4	a	a	DET
m-782	140	5	polycyclic	polycyclic	NOUN
m-782	140	6	group	group	NOUN
m-782	140	7	g.	g.	NOUN
m-782	141	1	we	we	PRON
m-782	141	2	aim	aim	VERB
m-782	141	3	to	to	PART
m-782	141	4	show	show	VERB
m-782	141	5	that	that	SCONJ
m-782	141	6	the	the	DET
m-782	141	7	pro	pro	ADJ
m-782	141	8	-	-	ADJ
m-782	141	9	nilpotent	nilpotent	ADJ
m-782	141	10	completion	completion	NOUN
m-782	141	11	of	of	ADP
m-782	141	12	g	g	NOUN
m-782	141	13	,	,	PUNCT
m-782	141	14	denoted	denote	VERB
m-782	141	15	�	�	PROPN
m-782	141	16	�	�	PROPN
m-782	141	17	,	,	PUNCT
m-782	141	18	with	with	ADP
m-782	141	19	respect	respect	NOUN
m-782	141	20	to	to	ADP
m-782	141	21	the	the	DET
m-782	141	22	pro	pro	ADJ
m-782	141	23	-	-	ADJ
m-782	141	24	nilpotent	nilpotent	ADJ
m-782	141	25	topology	topology	NOUN
m-782	141	26	is	be	AUX
m-782	141	27	locally	locally	ADV
m-782	141	28	polycyclic	polycyclic	NOUN
m-782	141	29	.	.	PUNCT
m-782	142	1	doi	doi	PROPN
m-782	142	2	10.5281	10.5281	NUM
m-782	142	3	/	/	SYM
m-782	142	4	zenodo.10511744	zenodo.10511744	PROPN
m-782	142	5	ijo	ijo	PROPN
m-782	142	6	journals	journal	NOUN
m-782	142	7	volume	volume	NOUN
m-782	142	8	07	07	NUM
m-782	143	1	|	|	NOUN
m-782	143	2	issue	issue	NOUN
m-782	143	3	01	01	NUM
m-782	144	1	|	|	CCONJ
m-782	144	2	january	january	PROPN
m-782	144	3	2024	2024	NUM
m-782	144	4	|	|	ADV
m-782	144	5	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-782	144	6	10	10	NUM
m-782	144	7	ijo	ijo	PROPN
m-782	144	8	international	international	PROPN
m-782	144	9	journal	journal	PROPN
m-782	144	10	of	of	ADP
m-782	144	11	mathematics	mathematics	PROPN
m-782	144	12	(	(	PUNCT
m-782	144	13	issn	issn	PROPN
m-782	144	14	:	:	PUNCT
m-782	144	15	2992	2992	NUM
m-782	144	16	-	-	SYM
m-782	144	17	4421	4421	NUM
m-782	144	18	)	)	PUNCT
m-782	145	1	michael	michael	PROPN
m-782	145	2	n.	n.	PROPN
m-782	145	3	john	john	PROPN
m-782	145	4	*	*	PROPN
m-782	145	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-782	145	6	volume	volume	NOUN
m-782	145	7	07	07	NUM
m-782	145	8	issue	issue	NOUN
m-782	145	9	01	01	NUM
m-782	145	10	||	||	NUM
m-782	145	11	january	january	PROPN
m-782	145	12	.	.	PUNCT
m-782	145	13	,	,	PUNCT
m-782	145	14	2024	2024	NUM
m-782	145	15	||	||	NOUN
m-782	145	16	recall	recall	NOUN
m-782	145	17	that	that	SCONJ
m-782	145	18	the	the	DET
m-782	145	19	pro	pro	ADJ
m-782	145	20	-	-	ADJ
m-782	145	21	nilpotent	nilpotent	ADJ
m-782	145	22	completion	completion	NOUN
m-782	145	23	�	�	PROPN
m-782	145	24	�	�	PROPN
m-782	145	25	is	be	AUX
m-782	145	26	constructed	construct	VERB
m-782	145	27	as	as	ADP
m-782	145	28	the	the	DET
m-782	145	29	inverse	inverse	NOUN
m-782	145	30	limit	limit	NOUN
m-782	145	31	of	of	ADP
m-782	145	32	the	the	DET
m-782	145	33	family	family	NOUN
m-782	145	34	of	of	ADP
m-782	145	35	all	all	DET
m-782	145	36	nilpotent	nilpotent	ADJ
m-782	145	37	quotients	quotient	NOUN
m-782	145	38	of	of	ADP
m-782	145	39	g.	g.	PROPN
m-782	145	40	specifically	specifically	ADV
m-782	145	41	,	,	PUNCT
m-782	145	42	if{ni	if{ni	PUNCT
m-782	145	43	}	}	PUNCT
m-782	145	44	is	be	AUX
m-782	145	45	the	the	DET
m-782	145	46	family	family	NOUN
m-782	145	47	of	of	ADP
m-782	145	48	all	all	DET
m-782	145	49	normal	normal	ADJ
m-782	145	50	nilpotent	nilpotent	ADJ
m-782	145	51	subgroups	subgroup	NOUN
m-782	145	52	of	of	ADP
m-782	145	53	g	g	NOUN
m-782	145	54	ordered	order	VERB
m-782	145	55	by	by	ADP
m-782	145	56	inclusion	inclusion	NOUN
m-782	145	57	,	,	PUNCT
m-782	145	58	then	then	ADV
m-782	145	59	�	�	PROPN
m-782	145	60	�	�	PROPN
m-782	145	61	=	=	SYM
m-782	145	62	lim	lim	PROPN
m-782	145	63	←	←	PROPN
m-782	145	64	�	�	PROPN
m-782	145	65	/	/	SYM
m-782	145	66	�	�	PROPN
m-782	145	67	�	�	PROPN
m-782	145	68	where	where	SCONJ
m-782	145	69	the	the	DET
m-782	145	70	morphisms	morphism	NOUN
m-782	145	71	in	in	ADP
m-782	145	72	the	the	DET
m-782	145	73	inverse	inverse	NOUN
m-782	145	74	limit	limit	NOUN
m-782	145	75	are	be	AUX
m-782	145	76	the	the	DET
m-782	145	77	natural	natural	ADJ
m-782	145	78	projection	projection	NOUN
m-782	145	79	maps	map	NOUN
m-782	145	80	.	.	PUNCT
m-782	146	1	since	since	SCONJ
m-782	146	2	g	g	PROPN
m-782	146	3	is	be	AUX
m-782	146	4	polycyclic	polycyclic	NOUN
m-782	146	5	,	,	PUNCT
m-782	146	6	it	it	PRON
m-782	146	7	has	have	VERB
m-782	146	8	a	a	DET
m-782	146	9	subnormal	subnormal	ADJ
m-782	146	10	series	series	NOUN
m-782	146	11	1	1	NUM
m-782	146	12	=	=	SYM
m-782	146	13	g0	g0	PROPN
m-782	146	14	⊴	⊴	ADP
m-782	146	15	g1	g1	PROPN
m-782	146	16	⊴	⊴	ADP
m-782	146	17	…	…	PUNCT
m-782	146	18	⊴	⊴	NUM
m-782	146	19	gk	gk	NOUN
m-782	146	20	=	=	SYM
m-782	146	21	g	g	PROPN
m-782	146	22	,	,	PUNCT
m-782	146	23	where	where	SCONJ
m-782	146	24	each	each	DET
m-782	146	25	factor	factor	NOUN
m-782	146	26	group	group	NOUN
m-782	146	27	gi+1	gi+1	PROPN
m-782	146	28	/	/	SYM
m-782	146	29	gi	gi	NOUN
m-782	146	30	is	be	AUX
m-782	146	31	cyclic	cyclic	ADJ
m-782	146	32	.	.	PUNCT
m-782	147	1	consider	consider	VERB
m-782	147	2	the	the	DET
m-782	147	3	corresponding	corresponding	ADJ
m-782	147	4	subnormal	subnormal	ADJ
m-782	147	5	series	series	NOUN
m-782	147	6	induced	induce	VERB
m-782	147	7	on	on	ADP
m-782	147	8	each	each	DET
m-782	147	9	g	g	PROPN
m-782	147	10	/	/	SYM
m-782	147	11	ni	ni	NOUN
m-782	147	12	:	:	PUNCT
m-782	147	13	1	1	NUM
m-782	147	14	=	=	PROPN
m-782	147	15	g0	g0	PROPN
m-782	147	16	/	/	SYM
m-782	147	17	ni	ni	PROPN
m-782	147	18	⊴	⊴	NOUN
m-782	147	19	g1	g1	PROPN
m-782	147	20	/	/	SYM
m-782	147	21	ni⊴	ni⊴	NOUN
m-782	147	22	…	…	SYM
m-782	147	23	⊴gk	⊴gk	NOUN
m-782	147	24	/	/	SYM
m-782	147	25	ni	ni	NOUN
m-782	147	26	=	=	PROPN
m-782	147	27	g	g	PROPN
m-782	147	28	/	/	SYM
m-782	147	29	ni	ni	PROPN
m-782	147	30	.	.	PROPN
m-782	148	1	since	since	SCONJ
m-782	148	2	each	each	DET
m-782	148	3	factor	factor	NOUN
m-782	148	4	groupgi+1	groupgi+1	PROPN
m-782	148	5	/	/	SYM
m-782	148	6	gi	gi	PROPN
m-782	148	7	is	be	AUX
m-782	148	8	cyclic	cyclic	ADJ
m-782	148	9	,	,	PUNCT
m-782	148	10	the	the	DET
m-782	148	11	corresponding	corresponding	ADJ
m-782	148	12	factor	factor	NOUN
m-782	148	13	groups	group	NOUN
m-782	148	14	(	(	PUNCT
m-782	148	15	gi+1	gi+1	NOUN
m-782	148	16	/	/	SYM
m-782	148	17	gi	gi	NOUN
m-782	148	18	)	)	PUNCT
m-782	148	19	/ni	/ni	PUNCT
m-782	148	20	are	be	AUX
m-782	148	21	also	also	ADV
m-782	148	22	cyclic	cyclic	ADJ
m-782	148	23	.	.	PUNCT
m-782	149	1	this	this	PRON
m-782	149	2	implies	imply	VERB
m-782	149	3	that	that	SCONJ
m-782	149	4	each	each	DET
m-782	149	5	g	g	PROPN
m-782	149	6	/	/	SYM
m-782	149	7	ni	ni	PROPN
m-782	149	8	is	be	AUX
m-782	149	9	a	a	DET
m-782	149	10	polycyclic	polycyclic	NOUN
m-782	149	11	group	group	NOUN
m-782	149	12	.	.	PUNCT
m-782	150	1	now	now	ADV
m-782	150	2	,	,	PUNCT
m-782	150	3	let	let	VERB
m-782	150	4	{	{	PUNCT
m-782	150	5	hj	hj	PART
m-782	150	6	}	}	PUNCT
m-782	150	7	be	be	AUX
m-782	150	8	the	the	DET
m-782	150	9	family	family	NOUN
m-782	150	10	of	of	ADP
m-782	150	11	all	all	DET
m-782	150	12	normal	normal	ADJ
m-782	150	13	subgroups	subgroup	NOUN
m-782	150	14	of	of	ADP
m-782	150	15	g	g	PROPN
m-782	150	16	that	that	PRON
m-782	150	17	are	be	AUX
m-782	150	18	contained	contain	VERB
m-782	150	19	in	in	ADP
m-782	150	20	some	some	DET
m-782	150	21	ni	ni	PROPN
m-782	150	22	.	.	PROPN
m-782	151	1	each	each	DET
m-782	151	2	hj	hj	PROPN
m-782	151	3	is	be	AUX
m-782	151	4	nilpotent	nilpotent	ADJ
m-782	151	5	because	because	SCONJ
m-782	151	6	it	it	PRON
m-782	151	7	is	be	AUX
m-782	151	8	contained	contain	VERB
m-782	151	9	in	in	ADP
m-782	151	10	a	a	DET
m-782	151	11	nilpotent	nilpotent	ADJ
m-782	151	12	subgroup	subgroup	NOUN
m-782	151	13	ni	ni	PROPN
m-782	151	14	.	.	PROPN
m-782	151	15	therefore	therefore	ADV
m-782	151	16	,	,	PUNCT
m-782	151	17	�	�	PROPN
m-782	151	18	�	�	PROPN
m-782	151	19	is	be	AUX
m-782	151	20	the	the	DET
m-782	151	21	inverse	inverse	ADJ
m-782	151	22	limit	limit	NOUN
m-782	151	23	of	of	ADP
m-782	151	24	polycyclic	polycyclic	NOUN
m-782	151	25	groups	group	NOUN
m-782	151	26	,	,	PUNCT
m-782	151	27	and	and	CCONJ
m-782	151	28	it	it	PRON
m-782	151	29	is	be	AUX
m-782	151	30	locally	locally	ADV
m-782	151	31	polycyclic	polycyclic	ADJ
m-782	151	32	.	.	PUNCT
m-782	152	1	thus	thus	ADV
m-782	152	2	,	,	PUNCT
m-782	152	3	we	we	PRON
m-782	152	4	have	have	AUX
m-782	152	5	shown	show	VERB
m-782	152	6	that	that	SCONJ
m-782	152	7	the	the	DET
m-782	152	8	pro	pro	ADJ
m-782	152	9	-	-	ADJ
m-782	152	10	nilpotent	nilpotent	ADJ
m-782	152	11	completion	completion	NOUN
m-782	152	12	�	�	NOUN
m-782	152	13	�	�	PROPN
m-782	152	14	of	of	ADP
m-782	152	15	a	a	DET
m-782	152	16	polycyclic	polycyclic	NOUN
m-782	152	17	group	group	NOUN
m-782	152	18	g	g	PROPN
m-782	152	19	is	be	AUX
m-782	152	20	locally	locally	ADV
m-782	152	21	polycyclic	polycyclic	ADJ
m-782	152	22	.	.	PUNCT
m-782	153	1	the	the	DET
m-782	153	2	theorem	theorem	NOUN
m-782	153	3	is	be	AUX
m-782	153	4	proved	prove	VERB
m-782	153	5	.	.	PUNCT
m-782	154	1	4.conclusion	4.conclusion	NUM
m-782	154	2	this	this	DET
m-782	154	3	research	research	NOUN
m-782	154	4	contributes	contribute	VERB
m-782	154	5	to	to	ADP
m-782	154	6	the	the	DET
m-782	154	7	understanding	understanding	NOUN
m-782	154	8	of	of	ADP
m-782	154	9	para-	para-	NOUN
m-782	154	10	�	�	PROPN
m-782	154	11	relations	relation	NOUN
m-782	154	12	and	and	CCONJ
m-782	154	13	their	their	PRON
m-782	154	14	implications	implication	NOUN
m-782	154	15	for	for	ADP
m-782	154	16	residually	residually	ADV
m-782	154	17	nilpotent	nilpotent	ADJ
m-782	154	18	groups	group	NOUN
m-782	154	19	.	.	PUNCT
m-782	155	1	the	the	DET
m-782	155	2	findings	finding	NOUN
m-782	155	3	shed	shed	VERB
m-782	155	4	light	light	NOUN
m-782	155	5	on	on	ADP
m-782	155	6	the	the	DET
m-782	155	7	interplay	interplay	NOUN
m-782	155	8	between	between	ADP
m-782	155	9	these	these	DET
m-782	155	10	groups	group	NOUN
m-782	155	11	,	,	PUNCT
m-782	155	12	providing	provide	VERB
m-782	155	13	insights	insight	NOUN
m-782	155	14	into	into	ADP
m-782	155	15	their	their	PRON
m-782	155	16	structural	structural	ADJ
m-782	155	17	properties	property	NOUN
m-782	155	18	,	,	PUNCT
m-782	155	19	doi	doi	X
m-782	155	20	10.5281	10.5281	NUM
m-782	155	21	/	/	SYM
m-782	155	22	zenodo.10511744	zenodo.10511744	PROPN
m-782	155	23	ijo	ijo	PROPN
m-782	155	24	journals	journal	NOUN
m-782	155	25	volume	volume	NOUN
m-782	155	26	07	07	NUM
m-782	155	27	|	|	NOUN
m-782	155	28	issue	issue	NOUN
m-782	155	29	01	01	NUM
m-782	156	1	|	|	CCONJ
m-782	156	2	january	january	PROPN
m-782	156	3	2024	2024	NUM
m-782	156	4	|	|	ADV
m-782	156	5	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-782	156	6	11	11	NUM
m-782	156	7	ijo	ijo	PROPN
m-782	156	8	international	international	PROPN
m-782	156	9	journal	journal	PROPN
m-782	156	10	of	of	ADP
m-782	156	11	mathematics	mathematics	PROPN
m-782	156	12	(	(	PUNCT
m-782	156	13	issn	issn	PROPN
m-782	156	14	:	:	PUNCT
m-782	156	15	2992	2992	NUM
m-782	156	16	-	-	SYM
m-782	156	17	4421	4421	NUM
m-782	156	18	)	)	PUNCT
m-782	157	1	michael	michael	PROPN
m-782	157	2	n.	n.	PROPN
m-782	157	3	john	john	PROPN
m-782	157	4	*	*	PROPN
m-782	157	5	http://ijojournals.com/	http://ijojournals.com/	PROPN
m-782	157	6	volume	volume	NOUN
m-782	157	7	07	07	NUM
m-782	157	8	issue	issue	NOUN
m-782	157	9	01	01	NUM
m-782	157	10	||	||	NUM
m-782	157	11	january	january	PROPN
m-782	157	12	.	.	PUNCT
m-782	157	13	,	,	PUNCT
m-782	157	14	2024	2024	NUM
m-782	157	15	||	||	NOUN
m-782	157	16	particularly	particularly	ADV
m-782	157	17	in	in	ADP
m-782	157	18	the	the	DET
m-782	157	19	context	context	NOUN
m-782	157	20	of	of	ADP
m-782	157	21	finitely	finitely	ADV
m-782	157	22	generated	generate	VERB
m-782	157	23	groups	group	NOUN
m-782	157	24	and	and	CCONJ
m-782	157	25	certain	certain	ADJ
m-782	157	26	polycyclic	polycyclic	ADJ
m-782	157	27	groups	group	NOUN
m-782	157	28	.	.	PUNCT
m-782	158	1	the	the	DET
m-782	158	2	established	establish	VERB
m-782	158	3	results	result	NOUN
m-782	158	4	open	open	VERB
m-782	158	5	avenues	avenue	NOUN
m-782	158	6	for	for	ADP
m-782	158	7	further	further	ADJ
m-782	158	8	exploration	exploration	NOUN
m-782	158	9	in	in	ADP
m-782	158	10	the	the	DET
m-782	158	11	broader	broad	ADJ
m-782	158	12	landscape	landscape	NOUN
m-782	158	13	of	of	ADP
m-782	158	14	group	group	NOUN
m-782	158	15	theory	theory	NOUN
m-782	158	16	.	.	PUNCT
m-782	159	1	5	5	X
m-782	159	2	.	.	X
m-782	159	3	corresponding	correspond	VERB
m-782	159	4	author	author	NOUN
m-782	159	5	michael	michael	PROPN
m-782	159	6	nsikan	nsikan	PROPN
m-782	159	7	john	john	PROPN
m-782	159	8	is	be	AUX
m-782	159	9	currently	currently	ADV
m-782	159	10	a	a	DET
m-782	159	11	phd	phd	NOUN
m-782	159	12	student	student	NOUN
m-782	159	13	of	of	ADP
m-782	159	14	mathematics	mathematic	NOUN
m-782	159	15	at	at	ADP
m-782	159	16	akwa	akwa	ADJ
m-782	159	17	ibom	ibom	ADJ
m-782	159	18	state	state	NOUN
m-782	159	19	university	university	NOUN
m-782	159	20	.	.	PUNCT
m-782	160	1	michael	michael	PROPN
m-782	160	2	does	do	AUX
m-782	160	3	research	research	NOUN
m-782	160	4	in	in	ADP
m-782	160	5	algebra	algebra	NOUN
m-782	160	6	;	;	PUNCT
m-782	160	7	group	group	NOUN
m-782	160	8	theory	theory	NOUN
m-782	160	9	,	,	PUNCT
m-782	160	10	computational	computational	ADJ
m-782	160	11	group	group	NOUN
m-782	160	12	theory	theory	NOUN
m-782	160	13	,	,	PUNCT
m-782	160	14	algebraic	algebraic	ADJ
m-782	160	15	cryptography	cryptography	NOUN
m-782	160	16	,	,	PUNCT
m-782	160	17	number	number	NOUN
m-782	160	18	theory	theory	NOUN
m-782	160	19	,	,	PUNCT
m-782	160	20	combinatorics	combinatorics	PROPN
m-782	160	21	,	,	PUNCT
m-782	160	22	blockchain	blockchain	PROPN
m-782	160	23	technology	technology	PROPN
m-782	160	24	.	.	PUNCT
m-782	161	1	for	for	ADP
m-782	161	2	more	more	ADJ
m-782	161	3	of	of	ADP
m-782	161	4	his	his	PRON
m-782	161	5	work	work	NOUN
m-782	161	6	,	,	PUNCT
m-782	161	7	read	read	VERB
m-782	161	8	from	from	ADP
m-782	161	9	[	[	X
m-782	161	10	5	5	NUM
m-782	161	11	]	]	PUNCT
m-782	161	12	to	to	ADP
m-782	161	13	[	[	X
m-782	161	14	31	31	NUM
m-782	161	15	]	]	PUNCT
m-782	161	16	references	reference	NOUN
m-782	161	17	[	[	X
m-782	161	18	1	1	NUM
m-782	161	19	]	]	PUNCT
m-782	161	20	hall	hall	NOUN
m-782	161	21	,	,	PUNCT
m-782	161	22	m.	m.	NOUN
m-782	161	23	(	(	PUNCT
m-782	161	24	2013	2013	NUM
m-782	161	25	)	)	PUNCT
m-782	161	26	.	.	PUNCT
m-782	162	1	theory	theory	NOUN
m-782	162	2	of	of	ADP
m-782	162	3	groups	group	NOUN
m-782	162	4	.	.	PUNCT
m-782	163	1	courier	courier	NOUN
m-782	163	2	corporation	corporation	NOUN
m-782	163	3	.	.	PUNCT
m-782	164	1	[	[	X
m-782	164	2	2	2	NUM
m-782	164	3	]	]	PUNCT
m-782	164	4	gruenberg	gruenberg	NOUN
m-782	164	5	,	,	PUNCT
m-782	164	6	k.	k.	PROPN
m-782	164	7	w.	w.	PROPN
m-782	164	8	(	(	PUNCT
m-782	164	9	1967	1967	NUM
m-782	164	10	)	)	PUNCT
m-782	164	11	.	.	PUNCT
m-782	165	1	cohomological	cohomological	ADJ
m-782	165	2	topics	topic	NOUN
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m-782	168	4	word	word	NOUN
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m-782	170	5	,	,	PUNCT
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m-782	171	3	.	.	PUNCT
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m-782	175	8	,	,	PUNCT
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m-782	175	13	10	10	NUM
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m-782	177	14	group	group	NOUN
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m-782	177	19	"	"	PUNCT
m-782	177	20	,	,	PUNCT
m-782	177	21	international	international	ADJ
m-782	177	22	journal	journal	NOUN
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m-782	177	25	research	research	NOUN
m-782	177	26	in	in	ADP
m-782	177	27	science	science	NOUN
m-782	177	28	,	,	PUNCT
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m-782	177	32	(	(	PUNCT
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m-782	177	40	-	-	SYM
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m-782	177	42	,	,	PUNCT
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m-782	177	45	:	:	PUNCT
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m-782	177	47	-	-	SYM
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m-782	177	49	,	,	PUNCT
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m-782	179	12	,	,	PUNCT
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m-782	179	14	,	,	PUNCT
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m-782	179	17	,	,	PUNCT
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m-782	179	25	of	of	ADP
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m-782	179	29	”	"	PUNCT
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m-782	181	1	|	|	CCONJ
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m-782	185	17	"	"	PUNCT
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m-782	185	45	.	.	PUNCT
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m-782	213	45	.	.	PUNCT
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