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(ISSN: 2992-4421 )                                                                                                                    Michael N. John* 

http://ijojournals.com/            Volume 07 Issue 01 || January., 2024 || 

 

 

Para-� Relations and Hirsch Length in Residually Nilpotent Groups 

 
 

Michael N. John 

Department of Mathematics, Akwa Ibom State University, Nigeria 
 

Stephen I. Okeke 

Department of Industrial Mathematics and Applied Statistics, David Umahi 

Federal University of Health Sciences, Uburu, Ebonyi State, Nigeria. 
 

Boniface O. Nwala 

Department of Mathematics, Ignatius Ajuru University of Education, Rumuolumeni, Port 

Harcourt, Rivers State, Nigeria 
 

Udoaka Otobong. G. 

Department of Mathematics, Akwa Ibom State University, Nigeria 

 
 

Abstract 

This research explores the interplay between residually nilpotent groups� and �, 

focusing on their relationship through the lens of para-� conditions and the 

Hirsch length. We establish criteria for � to be para-� concerning 

monomorphisms inducing isomorphisms between corresponding lower central 

quotients of � and �. Specifically, we investigate these conditions in the context 

of finitely generated residually nilpotent groups. Further, for certain polycyclic 

groups, we establish connections between para-� relations and the equality of 

Hirsch lengths. Additionally, we delve into the pro-nilpotent completions of these 

polycyclic groups, demonstrating their local polycyclic nature. 

 

KEYWORDS: Residually Nilpotent Groups, Para-� Relations, Hirsch Length, 

Lower Central Quotients, Pro-Nilpotent Completions, Polycyclic Groups. 

 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                    Michael N. John* 

http://ijojournals.com/            Volume 07 Issue 01 || January., 2024 || 

 

 

1. INTRODUCTION 

Residually nilpotent groups play a pivotal role in group theory, and understanding 

their relationships is essential for exploring the underlying algebraic structures. 

The study by [1] provides foundational insights into para-� conditions in group 

theory, particularly in the context of residually nilpotent groups. Hall's work lays 

the groundwork for understanding the interconnections between groups and the 

criteria for para-� relations.The concept of Hirsch length has been extensively 

explored in relation to finitely generated groups. [2]'s seminal work (1967) 

investigates the properties of the Hirsch length and its implications in the study of 

groups.The exploration of para-� relations within polycyclic groups is addressed 

by [3]. This work delves into the specific conditions and implications of para-� 

relations in the context of polycyclic structures.The study of pro-nilpotent 

completions in the realm of polycyclic groups is discussed by [4] and itprovides 

insights into the local polycyclic nature of these completions, contributing to the 

broader understanding of their properties. This research focuses on establishing 

and characterizing para-� relations between residually nilpotent groups� and �, 

with a particular emphasis on monomorphisms inducing isomorphisms between 

their lower central quotients. We extend our investigation to finitely generated 

groups and explore conditions for � to be para-�. Moreover, we explore the 

implications of para-� relations on the Hirsch length of certain polycyclic groups. 

2. PRELIMINARY 

Definition (Residually Nilpotent Groups) 2.1. A group G is said to be residually 

nilpotent if, for every non-identity element g in G, there exists a normal subgroup 

N of finite index such that N is a nilpotent group. In other words, every non-

identity element of the group can be separated from the identity by a finite-index 

normal subgroup that is nilpotent. 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                    Michael N. John* 

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Example (Residually Nilpotent Groups) 2.2. Consider the group G=Z×S3, 

where Z is the additive group of integers and S3 is the symmetric group on three 

elements. This group is a direct product of an infinite cyclic group (Z) and a finite 

group (S3). The group G is residually nilpotent because: 

1. For any non-identity element (n,e) ∈ G, where n is a non-zero integer and e 

is the identity element of S3, we can consider the subgroup N = {(0,e)}. 

This subgroup is of finite index, and N is nilpotent. 

2. For any non-identity element (0,σ) ∈ G, where σ is a non-identity 

permutation in S3, we can consider the subgroup N = { (0,σ), (0,e) }. This 

subgroup is of finite index, and N is nilpotent. 

Thus, G = Z × S3 is an example of a residually nilpotent group 

Definition (Para-� Relations) 2.3. Let G and H be two groups. The relation 

φ:G→H is a para-G relation if, for every normal subgroup N of G, the induced 

homomorphism φN:G/N→H/φ(N) is an isomorphism, where φ(N) = {φ(g)|g∈N} is 

the image of N under φ. 

In simpler terms, a para-G relation is a condition on a group homomorphism 

φ:G→H such that the homomorphism induces isomorphisms between 

corresponding lower central quotients for every normal subgroup of G. For a 

good homomorphism and the generators of its inner automorphism see [29] and 

[30]. 

 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                    Michael N. John* 

http://ijojournals.com/            Volume 07 Issue 01 || January., 2024 || 

 

 

Example (Para-� Relations) 2.4. Let's consider two groups G and H with the 

following properties: 

G = ⟨a, b| a2 = b2 = (ab)2 = e⟩ 

H = ⟨x,y| x2 = y2 = (xy)3 = e⟩ 

Define a group homomorphism φ:G→H by mapping a to x and b to y. This 

homomorphism φ is a para-G relation if, for every normal subgroup N of G, the 

induced homomorphism φN:G/N→H/φ(N) is an isomorphism. 

For example, consider the normal subgroup N = ⟨a⟩ of G. The induced 

homomorphism φN:G/N→H/φ(N) is an isomorphism because: 

φN(eN) = φ(e) = e = φ(N) 

φN(bN) = φ(b) = y = φ(N) 

This holds for every normal subgroup of G, and therefore, the homomorphism φ 

is a para-G relation between G and H. 

Definition (Hirsch Length) 2.5. The Hirsch length of a group G, denoted as 

h(G), is a non-negative integer that measures the growth rate of the lower central 

series of G. Specifically, h(G) is the length of the shortest possible generating 

tuple (g1,g2,…,gk) for G such that the i-th term of the lower central series of G is 

generated by g1,g2,…,gi for each i from 1 to k. 

In other words, h(G) is the smallest integer k such that G(k)={e}, where G(k) 

denotes the k-th term of the lower central series of G. 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                    Michael N. John* 

http://ijojournals.com/            Volume 07 Issue 01 || January., 2024 || 

 

 

Example (Hirsch Length) 2.6. Consider the free group F2 on two generators a 

and b, i.e., F2=⟨a,b|⟩. The lower central series of F2 is given by: 

F2
(1)=F2 

F2
(2) = [F2,F2] = ⟨[a,b]⟩ 

F2
(3) = [F2,F2

(2)] 

And so on. 

In this case, the Hirsch length h(F2) is 2 because the shortest generating tuple 

(g1,g2) is (a,[a,b]), and F2
(2) = ⟨[a,b]⟩ is generated by a and [a,b]. If one tries to 

generate F2
(3), a longer tuple is needed. 

So, for the free group F2,h(F2) = 2. 

Definition (Pro-Nilpotent Completions) 2.7. Let G be a group. The pro-

nilpotent completion of G, denoted as ��nil or ����, is the completion of G with 

respect to the pro-nilpotent topology. The pro-nilpotent topology on G is defined 

by the collection of all normal subgroups N of G such that the quotient G/N is 

nilpotent. 

The pro-nilpotent completion ��nil is the projective limit of the nilpotent quotients 

G/N over all normal subgroups N of G. Formally, it is given by: 

��nil = lim← �/� 

where the projective limit is taken over all normal subgroups N of G, and each 

G/N is a nilpotent group. 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                    Michael N. John* 

http://ijojournals.com/            Volume 07 Issue 01 || January., 2024 || 

 

 

Example (Pro-Nilpotent Completions) 2.8. Consider the additive group of 

integers Z. The pro-nilpotent completion ��nil is obtained by considering all normal 

subgroups N of Z such that the quotient Z/N is a nilpotent group. 

Since every quotient Z/nZ is nilpotent (as it is a cyclic group of prime order), the 

pro-nilpotent completion ��nil is the projective limit of all these nilpotent quotients: 

��nil= lim← �/�� 

This pro-nilpotent completion can be identified with the ring of p-adic integers Zp, 

where p is any prime number. The pro-nilpotent completion captures the p-adic 

topology of the integers. 

3.CENTRAL IDEA  

Lemma 3.1. Characterization of para-� relations in finitely generated residually 

nilpotent groups. 

Statement: Let G be a finitely generated residually nilpotent group. A group 

homomorphism φ:G→H is a para-� relation if and only if, for every finitely 

generated subgroup K of G, the kernel ker(φ↾K) is nilpotent. 

Proof: 

Forward Direction: Assume φ:G→H is a para-� relation. This implies that for 

every normal subgroup N of G, the induced homomorphism φN:G/N→H/φ(N) is 

an isomorphism. Consider a finitely generated subgroup K of G, and let L be a 

normal subgroup of K. Since K is finitely generated, L is also finitely generated. 

Now, consider the homomorphism φ↾K:K→H obtained by restricting φ to K. The 

kernel of φ↾K isker(φ↾K) = K∩ker(φ), where ker(φ) is the kernel of φ in G. 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                    Michael N. John* 

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Since φ is a para-� relation, ker(φ) is nilpotent. As L is a normal subgroup of K, L 

is also a normal subgroup of ker(φ). Thus, the quotient ker(φ)/L is nilpotent. By 

the correspondence theorem, this implies that (ker(φ)/L)∩K/L is nilpotent. 

Now, consider the homomorphism φK/L:K/L→H/φ(L) induced by φ on the 

quotient group K/L. The kernel of φK/L is (ker(φ)/L)∩K/L. Since this intersection is 

nilpotent, it follows that φK/L is an isomorphism. Therefore, φ↾K has a nilpotent 

kernel. 

Backward Direction: Conversely, assume that for every finitely generated 

subgroup K of G, the kernel ker(φ↾K) is nilpotent. We need to show that φ is a 

para-� relation. 

Let N be a normal subgroup of G, and consider the induced homomorphism φN

:G/N→H/φ(N). We aim to show that φN is an isomorphism. 

Take any finitely generated subgroup K/N of G/N. By the correspondence 

theorem, this corresponds to a finitely generated subgroup K of G containing N. 

Now, consider the homomorphism φK:K→H obtained by restricting φ to K. By 

assumption, the kernel ker(φK) = K∩ker(φ) is nilpotent. 

Let L be the normal subgroup L = K∩N. Since ker(φK) is nilpotent, it follows that 

(ker(φK)/L)∩(K/L) is nilpotent. Now, consider the homomorphism φK/L

:K/L→H/φ(L) induced by φ on the quotient group K/L. The kernel of φK/L is 

(ker(φK)/L)∩(K/L), which is nilpotent. 

Therefore, φK/L is an isomorphism. Since K/N was an arbitrary finitely generated 

subgroup of G/N, this holds for all finitely generated subgroups of G/N. Thus, φN 

is an isomorphism. 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                    Michael N. John* 

http://ijojournals.com/            Volume 07 Issue 01 || January., 2024 || 

 

 

Since N was an arbitrary normal subgroup of G, this establishes that φ is a para-

� relation. 

By proving both directions, we conclude that a group homomorphism φ:G→H is a 

para-� relation if and only if, for every finitely generated subgroup K of G, the 

kernel ker(φ↾K) is nilpotent. The lemma is proved. 

Proposition 3.2. Sufficient conditions on monomorphisms for � to be para-�. 

Statement: Let φ:G→H be a monomorphism, where G is a finitely generated 

residually nilpotent group, and H is a group. If, for every finitely generated 

subgroup K of G, the image φ(K) is a para-� relation in H, then H is para-�. 

Proof: 

Assume φ:G→H is a monomorphism, where G is finitely generated and 

residually nilpotent, and H is a group. Suppose that for every finitely generated 

subgroup K of G, the image φ(K) is a para-� relation in H. We aim to show that H 

is para-�. 

Let N be a normal subgroup of H, and consider the induced homomorphism φN

:G/ker(φ)→H/N. We need to show that φN is an isomorphism. 

Consider any finitely generated subgroup K/ker(φ) of G/ker(φ). By the 

correspondence theorem, this corresponds to a finitely generated subgroup K of 

G containing ker(φ). Now, the image φ(K) is a para-� relation in H, as per our 

assumption. 

Therefore, the induced homomorphism φK:K→H obtained by restricting φ to K is 

a para-� relation in H. This implies that the induced homomorphism φK/ker(φ)

:K/ker(φ)→φ(K) is an isomorphism. 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                    Michael N. John* 

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Now, consider the homomorphism φK/N:K/N→H/N induced by φ on the quotient 

group K/N. This is the composition of the isomorphism φK/ker(φ) and the natural 

projection K/ker(φ)→K/N. Since compositions of isomorphisms are 

isomorphisms, φK/N is an isomorphism. 

Since K/N was an arbitrary finitely generated subgroup of G/ker(φ), this holds for 

all finitely generated subgroups of G/ker(φ). Thus, φN is an isomorphism. 

Since N was an arbitrary normal subgroup of H, this establishes that H is para-�. 

By proving the sufficiency of the conditions on monomorphisms for H to be para-

�, the proposition is proved. 

Theorem 3.3. Implications of para-� relations on the Hirsch length of certain 

polycyclic groups. 

Statement: Let G be a finitely generated residually nilpotent group with a para-� 

relation in its subgroup H. If G is polycyclic, then the Hirsch length of G is 

bounded by the Hirsch length of H. 

Proof: 

Assume G is a finitely generated residually nilpotent group with a para-� relation 

in its subgroup H. Suppose G is polycyclic. We aim to show that the Hirsch 

length of G is bounded by the Hirsch length of H. 

Recall that the Hirsch length of a group is a measure of the growth rate of its 

lower central series. Let G = ⟨g1, g2, …, gn⟩ be a generating set for G. Since G is 

polycyclic, it has a subnormal series 

 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                    Michael N. John* 

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1 = G0 ⊴ G1 ⊴ …⊴ Gk = G, 

where each factor group Gi+1/Gi is cyclic. 

Consider the subgroup H′ = ⟨φ(g1), φ(g2),…,φ(gn)⟩ of H, where φ:G→H is the 

para-� relation. Since H is para-�, the Hirsch length of H is finite. 

Now, consider the induced homomorphism φi:Gi→H′ for each i=0,1,…,k. Since Gi 

is normal in Gi+1, the factor group Gi+1/Gi is cyclic, and φi(Gi+1) is cyclic in H′. 

Therefore, H′ also has a subnormal series 

1 = H0′ ⊴ H1′ ⊴ …⊴ Hk′ = H′, 

where each factor group H′i+1/Hi′ is cyclic. 

Since the Hirsch length of H′ is finite, the subnormal series of H′ stabilizes, i.e., 

there exists i0 such that Hi′ = H′i0 for all i≥i0. Correspondingly, the subnormal 

series of G stabilizes at i0, i.e., Gi = Gi0 for all i≥i0. 

This implies that the Hirsch length of G is bounded by the Hirsch length of H′, 

which is finite. Therefore, the theorem is proved. 

Theorem 3.4. Locally polycyclic nature of pro-nilpotent completions of specific 

polycyclic groups. 

Statement: Let G be a polycyclic group. The pro-nilpotent completion of G with 

respect to the pro-nilpotent topology is locally polycyclic. 

Proof: 

Consider a polycyclic group G. We aim to show that the pro-nilpotent completion 

of G, denoted��, with respect to the pro-nilpotent topology is locally polycyclic. 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                    Michael N. John* 

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Recall that the pro-nilpotent completion �� is constructed as the inverse limit of 

the family of all nilpotent quotients of G. Specifically, if{Ni} is the family of all 

normal nilpotent subgroups of G ordered by inclusion, then 

�� = lim
←

�/�� 

where the morphisms in the inverse limit are the natural projection maps. 

Since G is polycyclic, it has a subnormal series 

1 = G0 ⊴ G1 ⊴ …⊴ Gk = G, 

where each factor group Gi+1/Gi is cyclic. 

Consider the corresponding subnormal series induced on each G/Ni: 

1 = G0/Ni ⊴ G1/Ni⊴…⊴Gk/Ni = G/Ni. 

Since each factor groupGi+1/Gi is cyclic, the corresponding factor groups (Gi+1/Gi

)/Ni are also cyclic. This implies that each G/Ni is a polycyclic group. 

Now, let {Hj} be the family of all normal subgroups of G that are contained in 

some Ni. Each Hj is nilpotent because it is contained in a nilpotent subgroup Ni. 

Therefore, �� is the inverse limit of polycyclic groups, and it is locally polycyclic. 

Thus, we have shown that the pro-nilpotent completion �� of a polycyclic group G 

is locally polycyclic. The theorem is proved. 

4.CONCLUSION 

This research contributes to the understanding of para-� relations and their 

implications for residually nilpotent groups. The findings shed light on the 

interplay between these groups, providing insights into their structural properties, 

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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                    Michael N. John* 

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particularly in the context of finitely generated groups and certain polycyclic 

groups. The established results open avenues for further exploration in the 

broader landscape of group theory. 

5. CORRESPONDING AUTHOR 

Michael Nsikan John is currently a PhD student of Mathematics at Akwa Ibom 

State University. Michael does research in Algebra; Group theory, Computational 

Group theory, Algebraic Cryptography, Number theory, Combinatorics, 

Blockchain technology. For more of his work, read from [5] to [31] 

References 

[1] Hall, M. (2013). Theory of Groups. Courier Corporation. 

[2] Gruenberg, K. W. (1967). Cohomological Topics in Group Theory. Springer. 

[3] Robinson, D. J. S. (1996). Groups with solvable word problems. Walter de 

Gruyter. 

[4] Serre, J. P. (1997). Galois Cohomology. Springer. 

[5] Michael N. John &Udoaka O. G (2023). Algorithm and Cube-Lattice-Based 

Cryptography. International journal of Research Publication and reviews, Vol 

4, no 10, pp 3312-3315 October 2023. 

DOI:https://doi.org/10.55248/gengpi.4.1023.102842 

[6] Michael N. John, Udoaka O. G., "Computational Group Theory and Quantum-

Era Cryptography", International Journal of Scientific Research in Science, 

Engineering and Technology (IJSRSET), Online ISSN :2394-4099, Print ISSN 

:2395-1990, Volume 10 Issue 6,pp. 01-10, November-December 2023. 

Available at doi: https://doi.org/10.32628/IJSRSET2310556 

[7] Michael N. John, Udoaka, Otobong G., Alex Musa,"Key Agreement Protocol 

Using Conjugacy Classes of Finitely Generated group”, International Journal 

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https://scholar.google.com/citations?view_op=search_authors&hl=en&mauthors=label:combinatorics
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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                    Michael N. John* 

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of Scientific Research in Science and technology(IJSRST), Volume 10, Issue 

6, pp52-56. DOI: https://doi.org/10.32628/IJSRST2310645 

[8] Michael N. John, Udoaka, Otobong G., Boniface O. Nwala, "Elliptic-Curve 

Groups in Quantum-Era Cryptography”, ISAR Journal of science and 

technology, Volume 1, Issue 1, pp21-24. DOI: 

https://doi.org/10.5281/zenodo.10207536 

[9] Michael N John, UdoakaOtobong G and Alex Musa. Nilpotent groups in 

cryptographic key exchange protocol for N≥ 1. Journal of Mathematical 

Problems, Equations and Statistics. 2023; 4(2): 32-34. DOI: 

10.22271/math.2023.v4.i2a.103 

[10] Michael Nsikan John, UdoakaOtobong. G., & Alex Musa. (2023). 

SYMMETRIC BILINEAR CRYPTOGRAPHY ON ELLIPTIC CURVE AND LIE 

ALGEBRA. GPH - International Journal of Mathematics, 06(10), 01–15. 

https://doi.org/10.5281/zenodo.10200179 

[11] John, Michael N., Ozioma, O., Obi, P. N., Egbogho, H. E., & Udoaka, O. G. 

(2023). Lattices in Quantum-ERA Cryptography. International Journal of 

Research Publication and Reviews, V, 4(11), 2175–2179. 

https://doi.org/10.5281/zenodo.10207210 

[12] Michael N. John, OgoegbulemOzioma, UdoakaOtobong. G., Boniface O. 

Nwala, & Obi Perpetua Ngozi. (2023). CRYPTOGRAPHIC ENCRYPTION 

BASED ON RAIL-FENCE PERMUTATION CIPHER. GPH - International 

Journal of Mathematics, 06(11), 01–06. 

https://doi.org/10.5281/zenodo.10207316 

[13] Michael N. John, OgoegbulemOzioma, Obukohwo, Victor, & Henry 

EtarogheneEgbogho. (2023). NUMBER THEORY IN RSA ENCRYPTION 

SYSTEMS. GPH - International Journal of Mathematics, 06(11), 07–16. 

https://doi.org/10.5281/zenodo.10207361 

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https://doi.org/10.32628/IJSRST2310645
https://doi.org/10.5281/zenodo.10207536
https://doi.org/10.5281/zenodo.10200179
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https://doi.org/10.5281/zenodo.10207361


IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                    Michael N. John* 

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[14] John Michael. N., Bassey E. E., Udoaka O.G., Otobong J. T and Promise 

O.U (2023) On Finding the Number of Homomorphism from Q8 , International 

Journal of Mathematics and Statistics Studies, 11 (4), 20-26. doi: 

https://doi.org/10.37745/ijmss.13/vol11n42026 

[15] Michael N. John, Otobong G. Udoaka, &Itoro U. Udoakpan. (2023). Group 

Theory in Lattice-Based Cryptography. International Journal of Mathematics 

And Its Applications, 11(4), 111–125. Retrieved from 

https://ijmaa.in/index.php/ijmaa/article/view/1438 

[16] Michael N. John and Udoakpan I. U (2023) Fuzzy Group Action on an R-

Subgroup in a Near-Ring, International Journal of Mathematics and Statistics 

Studies, 11 (4), 27-31. Retrieved from https://eajournals.org/ijmss/wp-

content/uploads/sites/71/2023/12/Fuzzy-Group.pdf        DOI; 

https://doi.org/10.37745/ijmss.13/vol11n42731 

[17] Michael N. John, Edet, Effiong, &Otobong G. Udoaka. (2023). On Finding B-

Algebras Generated By Modulo Integer Groups �n. International Journal of 

Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 

2321 - 4759, Volume 11 Issue 6 || Nov. – Dec., 2023 || PP 01-04. Retrieved 

from https://www.ijmsi.org/Papers/Volume.11.Issue.6/11060104.pdf 

[18] Michael N. J., Ochonogor N., Ogoegbulem O. and Udoaka, O. G. (2023) 

Graph of Co-Maximal Subgroups in The Integer Modulo N Group, 

International Journal of Mathematics and Statistics Studies, 11 (4), 45-50. 

Retrieved from https://eajournals.org/ijmss/wp-

content/uploads/sites/71/2023/12/Graph-of-Co-Maximal-Subgroups.pdf  

DOI; https://doi.org/10.37745/ijmss.13/vol11n44550 

[19] Michael N. John, Otobong G. Udoaka & Alex Musa. (2023). Solvable Groups 

With Monomial Characters Of Prime Power Codegree And Monolithic 

Characters. BULLETIN OF MATHEMATICS AND STATISTICS RESEARCH: 

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(ISSN: 2992-4421 )                                                                                                                    Michael N. John* 

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[20] Michael N. J, Musa A., and Udoaka O.G. (2023) Conjugacy Classes in 

Finitely Generated Groups with Small Cancellation Properties, European 

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[21] Michael N. J., Ochonogor N., Ogoegbulem O. and Udoaka O. G. (2023), 

Modularity in Finite Groups: Characterizing Groups with Modular �- 

Subnormal Subgroups, International Journal of Mathematics and Computer 

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[22] John, M. N., Bassey, E. E., Godswill, I. C., & G., U. (2023). On The Structure 

and Classification of Finite Linear Groups: A Focus on Hall Classes and 

Nilpotency. International Journal Of Mathematics And Computer 

Research, 11(12), 3919-3925. https://doi.org/10.47191/ijmcr/v11i12.07 

[23] John, M. N., & U., U. I. (2023). On Strongly Base-Two Finite Groups with 

Trivial Frattini Subgroup: Conjugacy Classes and Core-Free 

Subgroup. International Journal Of Mathematics And Computer 

Research, 11(12), 3926-3932. https://doi.org/10.47191/ijmcr/v11i12.08 

[24] John, M. N., Etim, U. J,,&Udoaka O. G. (2023). Algebraic Structures and 

Applications: From Transformation Semigroups to Cryptography, Blockchain, 

and Computational Mathematics.  International Journal of Computer Science 

and Mathematical Theory (IJCSMT) E-ISSN 2545-5699 P-ISSN 2695-1924 

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http://www.bomsr.com/11.4.23/98-102 MICHAEL N. JOHN.pdf
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IJO - INTERNATIONAL JOURNAL OF MATHEMATICS  
(ISSN: 2992-4421 )                                                                                                                    Michael N. John* 

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[25] John, M. N., Ogoegbulem O., Etim, U. J,,&Udoaka O. G. (2023). 

Characterization Theorems for Just Infinite Profinite Residually Solvable Lie 

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[26] John, M. N., &Otobong. G, U. (2023). Algebraic and Topological Analysis of 

Enveloping Semigroups in Transformation Groups: Proximal Equivalence and 

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4421 ), 6(12), 09-23. DOI; https://doi.org/10.5281/zenodo.10443958 

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Application to Symmetric Cryptography. International Journal of Pure 

Mathematics DOI: 10.46300/91019.2022.9.13.  

[28] Udoaka O. G, Asibong-Ibe U. I. & David E. E. (2016). Rank ofproduct of 

certain algebraic classes. IOSR Journal of Mathematics, 12, e-ISSN: 2278-

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[29] Ndubuisi, O G Udoaka, K P Shum, and R B Abubakar, (2019). On 

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pp. 4793-4797, Online ISSN: 1920-3853; Print ISSN: 1715- 9997. 

[30] Udoaka, O. G. (2022). Generators and inner automorphism. THE 

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