id	sid	eid	entity	type
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m-782	2	6	michael n. john department	PERSON
m-782	2	7	akwa ibom state university	ORG
m-782	2	8	nigeria	GPE
m-782	2	9	stephen i.	PERSON
m-782	2	10	david umahi	PERSON
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m-782	2	12	uburu	GPE
m-782	2	13	nigeria	GPE
m-782	3	1	boniface o. nwala department	PERSON
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m-782	3	3	harcourt	GPE
m-782	3	4	nigeria	GPE
m-782	4	1	akwa ibom state university, nigeria	ORG
m-782	4	2	hirsch	PERSON
m-782	7	1	hirsch	PERSON
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m-782	10	3	01	DATE
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m-782	12	4	2024	CARDINAL
m-782	13	1	1	CARDINAL
m-782	14	1	hirsch	PERSON
m-782	15	1	2	CARDINAL
m-782	15	2	1967	DATE
m-782	15	3	hirsch	PERSON
m-782	15	4	3	CARDINAL
m-782	16	1	4	CARDINAL
m-782	19	1	hirsch	PERSON
m-782	20	1	2.	CARDINAL
m-782	20	2	2.1	CARDINAL
m-782	23	1	10.5281	CARDINAL
m-782	23	2	07	CARDINAL
m-782	24	1	01	DATE
m-782	25	1	january 2024	DATE
m-782	25	2	2	CARDINAL
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m-782	26	2	07	CARDINAL
m-782	26	3	01	CARDINAL
m-782	26	4	2024	CARDINAL
m-782	26	5	2.2	CARDINAL
m-782	27	1	three	CARDINAL
m-782	29	1	1	CARDINAL
m-782	31	1	2.	CARDINAL
m-782	31	2	σ	ORG
m-782	33	1	2.3	CARDINAL
m-782	34	1	two	CARDINAL
m-782	35	1	n under φ	ORG
m-782	36	1	φ	PERSON
m-782	36	2	29	CARDINAL
m-782	36	3	30	CARDINAL
m-782	37	1	10.5281	CARDINAL
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m-782	38	1	01	DATE
m-782	39	1	january 2024	DATE
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m-782	41	1	two	CARDINAL
m-782	41	2	ab)2	NORP
m-782	43	1	xy)3 = e⟩	ORG
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m-782	46	1	φ	PERSON
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m-782	47	1	hirsch	PERSON
m-782	47	2	1	CARDINAL
m-782	47	3	integer k	PERSON
m-782	47	4	07	CARDINAL
m-782	48	1	01	DATE
m-782	49	1	january 2024	DATE
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m-782	50	1	michael n. john* http://ijojournals.com/	PERSON
m-782	50	2	07	CARDINAL
m-782	50	3	01	CARDINAL
m-782	50	4	2024	CARDINAL
m-782	50	5	hirsch	PERSON
m-782	50	6	2.6	CARDINAL
m-782	51	1	two	CARDINAL
m-782	51	2	f2=⟨a	ORG
m-782	51	3	b|⟩.	NORP
m-782	51	4	f2	ORG
m-782	51	5	f2	PERSON
m-782	51	6	2	CARDINAL
m-782	52	1	b]⟩ f2	PERSON
m-782	52	2	3	CARDINAL
m-782	53	1	2	CARDINAL
m-782	54	1	hirsch	PERSON
m-782	54	2	2	CARDINAL
m-782	54	3	f2	ORG
m-782	54	4	2	CARDINAL
m-782	55	1	one	CARDINAL
m-782	55	2	3	CARDINAL
m-782	56	1	f2,h(f2	CARDINAL
m-782	56	2	2	CARDINAL
m-782	56	3	2.7	CARDINAL
m-782	61	1	10.5281	CARDINAL
m-782	61	2	07	CARDINAL
m-782	62	1	01	DATE
m-782	63	1	january 2024	DATE
m-782	63	2	5	CARDINAL
m-782	63	3	ijo international journal of mathematics	ORG
m-782	63	4	2992-4421	DATE
m-782	64	1	michael n. john* http://ijojournals.com/	PERSON
m-782	64	2	07	CARDINAL
m-782	64	3	01	CARDINAL
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m-782	64	5	2.8	CARDINAL
m-782	65	1	integers z.	PERSON
m-782	67	1	zp	ORG
m-782	69	1	3.central	CARDINAL
m-782	69	2	3.1	CARDINAL
m-782	71	1	φ	PERSON
m-782	72	1	φ	PERSON
m-782	75	1	φ	PERSON
m-782	75	2	φ	PERSON
m-782	75	3	φ	GPE
m-782	75	4	φ	ORG
m-782	75	5	10.5281	CARDINAL
m-782	75	6	07	CARDINAL
m-782	76	1	01	DATE
m-782	77	1	january 2024	DATE
m-782	77	2	6	CARDINAL
m-782	77	3	2992-4421	DATE
m-782	78	1	michael n. john* http://ijojournals.com/	PERSON
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m-782	78	3	01	CARDINAL
m-782	78	4	2024	CARDINAL
m-782	78	5	φ	ORG
m-782	79	1	k, l	ORG
m-782	81	1	ker(φ)/l)∩k/l	ORG
m-782	82	1	φ	ORG
m-782	82	2	k/l.	ORG
m-782	82	3	φk/l	ORG
m-782	82	4	ker(φ)/l)∩k	NORP
m-782	83	1	φ	PERSON
m-782	85	1	φ	ORG
m-782	88	1	k/n of g/n.	ORG
m-782	88	2	φ	PERSON
m-782	89	1	l = k∩n	PERSON
m-782	91	1	φ	ORG
m-782	91	2	k/l.	ORG
m-782	91	3	φk/l	ORG
m-782	93	1	k/n	FAC
m-782	93	2	g/n	ORG
m-782	94	1	10.5281	CARDINAL
m-782	94	2	07	CARDINAL
m-782	95	1	01	DATE
m-782	96	1	january 2024	DATE
m-782	96	2	7 ijo international journal of mathematics	ORG
m-782	96	3	2992-4421	DATE
m-782	97	1	michael n. john* http://ijojournals.com/	PERSON
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m-782	97	3	01	CARDINAL
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m-782	97	5	φ	ORG
m-782	98	1	φ	PERSON
m-782	99	1	lemma	ORG
m-782	100	1	3.2	CARDINAL
m-782	102	1	φ(k	PERSON
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m-782	104	1	φ(k	PERSON
m-782	106	1	k/ker(φ	ORG
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m-782	108	1	φ(k	PERSON
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m-782	110	1	10.5281	CARDINAL
m-782	110	2	07	CARDINAL
m-782	111	1	01	DATE
m-782	112	1	january 2024	DATE
m-782	112	2	8	CARDINAL
m-782	112	3	2992-4421	DATE
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m-782	113	3	01	CARDINAL
m-782	113	4	2024	CARDINAL
m-782	113	5	φ	ORG
m-782	113	6	k/n.	ORG
m-782	114	1	k/n	FAC
m-782	114	2	g/ker(φ	ORG
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m-782	117	1	3.3	CARDINAL
m-782	118	1	hirsch	PERSON
m-782	120	1	hirsch	PERSON
m-782	120	2	hirsch	PERSON
m-782	120	3	g	PERSON
m-782	121	1	hirsch	PERSON
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m-782	121	3	hirsch	PERSON
m-782	122	1	10.5281	CARDINAL
m-782	122	2	07	CARDINAL
m-782	122	3	01	DATE
m-782	123	1	january 2024	DATE
m-782	123	2	9	CARDINAL
m-782	123	3	ijo international journal of mathematics	ORG
m-782	123	4	2992-4421	DATE
m-782	124	1	michael n. john* http://ijojournals.com/	PERSON
m-782	124	2	07	CARDINAL
m-782	124	3	01	CARDINAL
m-782	125	1	2024	CARDINAL
m-782	125	2	g1	ORG
m-782	125	3	gk = g	PERSON
m-782	126	1	h′	PERSON
m-782	126	2	φ	PERSON
m-782	127	1	hirsch	PERSON
m-782	129	1	h′	PERSON
m-782	129	2	1	CARDINAL
m-782	130	1	h0′	PERSON
m-782	130	2	h′	PERSON
m-782	131	1	hirsch	PERSON
m-782	131	2	h′	ORG
m-782	131	3	h′	ORG
m-782	131	4	i0	ORG
m-782	133	1	i0	ORG
m-782	134	1	hirsch	PERSON
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m-782	134	3	h′	ORG
m-782	136	1	3.4	CARDINAL
m-782	142	1	10.5281	CARDINAL
m-782	142	2	07	CARDINAL
m-782	143	1	01	DATE
m-782	144	1	january 2024	DATE
m-782	144	2	10	CARDINAL
m-782	144	3	ijo international journal of mathematics	ORG
m-782	144	4	2992-4421	DATE
m-782	145	1	michael n. john* http://ijojournals.com/	PERSON
m-782	145	2	07	CARDINAL
m-782	145	3	01	CARDINAL
m-782	145	4	2024	CARDINAL
m-782	146	1	1	CARDINAL
m-782	146	2	g1	ORG
m-782	146	3	gk = g	PERSON
m-782	147	1	1	CARDINAL
m-782	147	2	g1/ni⊴…⊴gk	ORG
m-782	154	1	4.conclusion	CARDINAL
m-782	155	1	10.5281	CARDINAL
m-782	155	2	07	CARDINAL
m-782	155	3	01	DATE
m-782	156	1	january 2024	DATE
m-782	156	2	11	CARDINAL
m-782	156	3	ijo international journal of mathematics	ORG
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m-782	157	1	michael n. john* http://ijojournals.com/	PERSON
m-782	157	2	07	CARDINAL
m-782	157	3	01	CARDINAL
m-782	157	4	2024	CARDINAL
m-782	159	1	5	CARDINAL
m-782	159	2	michael nsikan john	PERSON
m-782	159	3	akwa ibom state university	ORG
m-782	160	1	michael	PERSON
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m-782	161	1	5	CARDINAL
m-782	161	2	31	CARDINAL
m-782	161	3	1	CARDINAL
m-782	161	4	2013	DATE
m-782	164	1	2	CARDINAL
m-782	164	2	k. w.	PERSON
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m-782	167	1	3	CARDINAL
m-782	167	2	d. j. s.	PERSON
m-782	167	3	1996	DATE
m-782	169	1	walter de gruyter	PERSON
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m-782	170	2	j. p.	PERSON
m-782	170	3	1997	DATE
m-782	171	1	galois	PERSON
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m-782	173	2	michael n. john &udoaka o. g	PERSON
m-782	173	3	2023	CARDINAL
m-782	175	1	international journal of research	ORG
m-782	175	2	4	CARDINAL
m-782	175	3	3312-3315	DATE
m-782	175	4	2023	CARDINAL
m-782	177	1	6	CARDINAL
m-782	177	2	michael n. john	PERSON
m-782	177	3	udoaka o. g.	PERSON
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m-782	177	5	10	CARDINAL
m-782	177	6	6,pp	CARDINAL
m-782	178	1	01-10	DATE
m-782	178	2	november-december 2023	DATE
m-782	179	1	7	CARDINAL
m-782	179	2	michael n. john	PERSON
m-782	179	3	otobong g.	PERSON
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m-782	180	1	01	DATE
m-782	181	1	january 2024	DATE
m-782	182	1	12	CARDINAL
m-782	182	2	https://doi.org/10.32628/ijsrset2310556 ijo international journal of mathematics	ORG
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m-782	183	5	10	CARDINAL
m-782	183	6	6	CARDINAL
m-782	183	7	pp52-56	PRODUCT
m-782	185	1	8	CARDINAL
m-782	185	2	michael n. john	PERSON
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m-782	185	7	1	CARDINAL
m-782	185	8	pp21-24	PRODUCT
m-782	185	9	9	CARDINAL
m-782	185	10	michael n john	PERSON
m-782	185	11	udoakaotobong g	PERSON
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m-782	186	1	1	CARDINAL
m-782	187	1	2023	CARDINAL
m-782	187	2	4(2	DATE
m-782	187	3	32	CARDINAL
m-782	187	4	10.22271	CARDINAL
m-782	187	5	10	CARDINAL
m-782	187	6	michael nsikan john	PERSON
m-782	187	7	udoakaotobong.	PERSON
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m-782	189	1	2023	CARDINAL
m-782	191	1	gph international journal of mathematics	ORG
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m-782	193	2	john	PERSON
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m-782	195	2	4(11	CARDINAL
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m-782	201	1	gph international journal of mathematics	ORG
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m-782	203	2	michael n. john	PERSON
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m-782	205	1	rsa	ORG
m-782	206	1	gph international journal of mathematics	ORG
m-782	206	2	06(11	CARDINAL
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m-782	207	2	07	CARDINAL
m-782	208	1	01	DATE
m-782	209	1	january 2024	DATE
m-782	210	1	13	CARDINAL
m-782	210	2	https://doi.org/10.5281/zenodo.10200179 https://doi.org/10.5281/zenodo.10207210 https://doi.org/10.5281/zenodo.10207316 https://doi.org/10.5281/zenodo.10207361 ijo international journal of mathematics	ORG
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m-782	213	4	2023	CARDINAL
m-782	213	5	q8	GPE
m-782	213	6	11	CARDINAL
m-782	213	7	4	CARDINAL
m-782	213	8	20-26	CARDINAL
m-782	215	1	15	CARDINAL
m-782	215	2	michael n. john	PERSON
m-782	215	3	otobong g. udoaka	PERSON
m-782	216	1	2023	CARDINAL
m-782	218	1	11(4	DATE
m-782	218	2	111–125	CARDINAL
m-782	220	1	16	CARDINAL
m-782	220	2	michael n. john	PERSON
m-782	220	3	i. u	PERSON
m-782	220	4	2023	CARDINAL
m-782	220	5	11	CARDINAL
m-782	220	6	4	CARDINAL
m-782	220	7	27-31	DATE
m-782	222	1	17	CARDINAL
m-782	222	2	michael n. john	PERSON
m-782	222	3	effiong, &	ORG
m-782	223	1	2023	CARDINAL
m-782	224	1	2321	CARDINAL
m-782	224	2	4767	CARDINAL
m-782	224	3	2321 4759	DATE
m-782	224	4	11	CARDINAL
m-782	224	5	6 || nov	QUANTITY
m-782	224	6	dec.	GPE
m-782	224	7	2023 ||	QUANTITY
m-782	225	1	01-04	CARDINAL
m-782	226	1	18	CARDINAL
m-782	226	2	michael n. j.	PERSON
m-782	226	3	ochonogor n.	PERSON
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m-782	226	6	11	CARDINAL
m-782	226	7	4	CARDINAL
m-782	226	8	45-50	CARDINAL
m-782	228	1	19	CARDINAL
m-782	228	2	michael n. john	PERSON
m-782	228	3	alex musa	PERSON
m-782	229	1	2023	CARDINAL
m-782	231	1	10.5281	CARDINAL
m-782	231	2	07	CARDINAL
m-782	231	3	01	DATE
m-782	232	1	january 2024	DATE
m-782	232	2	14 https://doi.org/10.37745/ijmss.13/vol11n42026 https://ijmaa.in/index.php/ijmaa/article/view/1438	QUANTITY
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m-782	233	5	98 102	CARDINAL
m-782	233	6	11	CARDINAL
m-782	233	7	7 ||	QUANTITY
m-782	233	8	dec.	GPE
m-782	233	9	2023 ||	QUANTITY
m-782	234	1	01-04	CARDINAL
m-782	235	1	20	CARDINAL
m-782	235	2	michael n. j	PERSON
m-782	235	3	musa a.	PERSON
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m-782	236	2	european journal of statistics	ORG
m-782	236	3	12	CARDINAL
m-782	236	4	1	CARDINAL
m-782	236	5	1	CARDINAL
m-782	237	1	21	CARDINAL
m-782	237	2	michael n. j.	PERSON
m-782	237	3	ochonogor n.	PERSON
m-782	237	4	2023	CARDINAL
m-782	237	5	11	CARDINAL
m-782	237	6	12	CARDINAL
m-782	237	7	3914	CARDINAL
m-782	238	1	https://doi.org/10.47191/ijmcr/v11i12.06	DATE
m-782	238	2	22	CARDINAL
m-782	238	3	m. n.	PERSON
m-782	238	4	e. e.	PERSON
m-782	238	5	i. c., & g.	PERSON
m-782	238	6	2023	CARDINAL
m-782	240	1	11(12	CARDINAL
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