IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (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. DOI 10.5281/zenodo.10511744 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | http://ijojournals.com/index.php/m/index 1 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. DOI 10.5281/zenodo.10511744 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | http://ijojournals.com/index.php/m/index 2 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Michael N. John* http://ijojournals.com/ Volume 07 Issue 01 || January., 2024 || 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]. DOI 10.5281/zenodo.10511744 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | http://ijojournals.com/index.php/m/index 3 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. DOI 10.5281/zenodo.10511744 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | http://ijojournals.com/index.php/m/index 4 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. DOI 10.5281/zenodo.10511744 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | http://ijojournals.com/index.php/m/index 5 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. DOI 10.5281/zenodo.10511744 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | http://ijojournals.com/index.php/m/index 6 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Michael N. John* http://ijojournals.com/ Volume 07 Issue 01 || January., 2024 || 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. DOI 10.5281/zenodo.10511744 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | http://ijojournals.com/index.php/m/index 7 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. DOI 10.5281/zenodo.10511744 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | http://ijojournals.com/index.php/m/index 8 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Michael N. John* http://ijojournals.com/ Volume 07 Issue 01 || January., 2024 || 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 DOI 10.5281/zenodo.10511744 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | http://ijojournals.com/index.php/m/index 9 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Michael N. John* http://ijojournals.com/ Volume 07 Issue 01 || January., 2024 || 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. DOI 10.5281/zenodo.10511744 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | http://ijojournals.com/index.php/m/index 10 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Michael N. John* http://ijojournals.com/ Volume 07 Issue 01 || January., 2024 || 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, DOI 10.5281/zenodo.10511744 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | http://ijojournals.com/index.php/m/index 11 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Michael N. John* http://ijojournals.com/ Volume 07 Issue 01 || January., 2024 || 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 DOI 10.5281/zenodo.10511744 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | http://ijojournals.com/index.php/m/index 12 https://scholar.google.com/citations?view_op=search_authors&hl=en&mauthors=label:combinatorics https://doi.org/10.55248/gengpi.4.1023.102842 https://doi.org/10.32628/IJSRSET2310556 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Michael N. John* http://ijojournals.com/ Volume 07 Issue 01 || January., 2024 || 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 DOI 10.5281/zenodo.10511744 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | http://ijojournals.com/index.php/m/index 13 https://doi.org/10.32628/IJSRST2310645 https://doi.org/10.5281/zenodo.10207536 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 (ISSN: 2992-4421 ) Michael N. John* http://ijojournals.com/ Volume 07 Issue 01 || January., 2024 || [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: DOI 10.5281/zenodo.10511744 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | http://ijojournals.com/index.php/m/index 14 https://doi.org/10.37745/ijmss.13/vol11n42026 https://ijmaa.in/index.php/ijmaa/article/view/1438 https://eajournals.org/ijmss/wp-content/uploads/sites/71/2023/12/Fuzzy-Group.pdf https://eajournals.org/ijmss/wp-content/uploads/sites/71/2023/12/Fuzzy-Group.pdf https://doi.org/10.37745/ijmss.13/vol11n42731 https://www.ijmsi.org/Papers/Volume.11.Issue.6/11060104.pdf https://eajournals.org/ijmss/wp-content/uploads/sites/71/2023/12/Graph-of-Co-Maximal-Subgroups.pdf https://eajournals.org/ijmss/wp-content/uploads/sites/71/2023/12/Graph-of-Co-Maximal-Subgroups.pdf https://doi.org/10.37745/ijmss.13/vol11n44550 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Michael N. John* http://ijojournals.com/ Volume 07 Issue 01 || January., 2024 || 98 - 102, Volume 11 Issue 7 || Oct. – Dec., 2023 || PP 01-04. Retrieved from http://www.bomsr.com/11.4.23/98- 102%20MICHAEL%20N.%20JOHN.pdfDOI:10.33329/bomsr.11.4.98 [20] Michael N. J, Musa A., and Udoaka O.G. (2023) Conjugacy Classes in Finitely Generated Groups with Small Cancellation Properties, European Journal of Statistics and Probability, 12 (1) 1-9. DOI: https://doi.org/10.37745/ejsp.2013/vol12n119 [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 Reserach, Volume 11 (12), 3914-3918. Retrieved from https://ijmcr.in/index.php/ijmcr/article/view/672/561 DOI; https://doi.org/10.47191/ijmcr/v11i12.06 [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 Vol 9. No.5 2023. DOI: https://doi.org/10.56201/ijcsmt.v9.no5.2023.pg82.101 DOI 10.5281/zenodo.10511744 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | http://ijojournals.com/index.php/m/index 15 http://www.bomsr.com/11.4.23/98-102 MICHAEL N. JOHN.pdf http://www.bomsr.com/11.4.23/98-102 MICHAEL N. JOHN.pdf http://www.bomsr.com/11.4.23/98-102 MICHAEL N. JOHN.pdf https://doi.org/10.37745/ejsp.2013/vol12n119 https://ijmcr.in/index.php/ijmcr/article/view/672/561 https://doi.org/10.47191/ijmcr/v11i12.06 https://doi.org/10.47191/ijmcr/v11i12.07 https://doi.org/10.47191/ijmcr/v11i12.08 https://doi.org/10.56201/ijcsmt.v9.no5.2023.pg82.101 IJO - INTERNATIONAL JOURNAL OF MATHEMATICS (ISSN: 2992-4421 ) Michael N. John* http://ijojournals.com/ Volume 07 Issue 01 || January., 2024 || [25] John, M. N., Ogoegbulem O., Etim, U. J,,&Udoaka O. G. (2023). Characterization Theorems for Just Infinite Profinite Residually Solvable Lie Algebras. International Journal of Computer Science and Mathematical Theory (IJCSMT) E-ISSN 2545-5699 P-ISSN 2695-1924 Vol 9. No.5 2023. DOI: https://doi.org/10.56201/ijcsmt.v9.no5.2023.pg102.113 [26] John, M. N., &Otobong. G, U. (2023). Algebraic and Topological Analysis of Enveloping Semigroups in Transformation Groups: Proximal Equivalence and Homomorphic Image. IJO - International Journal of Mathematics (ISSN: 2992- 4421 ), 6(12), 09-23. DOI; https://doi.org/10.5281/zenodo.10443958 [27] Udoaka O. G. & Frank E. A.,(2022). Finite Semi-group Modulo and Its 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- 5728, 6, ver. 1,pg 123-125. [29] Ndubuisi, O G Udoaka, K P Shum, and R B Abubakar, (2019). On Homomorphisms (Good Homomorphisms) Between Completely J^∘-Simple Semigroups Canadian Journal of Pure and Applied Sciences, Vol. 13, No. 2, pp. 4793-4797, Online ISSN: 1920-3853; Print ISSN: 1715- 9997. [30] Udoaka, O. G. (2022). Generators and inner automorphism. THE COLLOQUIUM -A Multidisciplinary Thematc Policy Journal www.ccsonlinejournals.com. Volume 10, Number 1 , Pages 102 -111 CC-BY- NC-SA 4.0 International Print ISSN : 2971-6624 eISSN: 2971-6632. [31] Udoaka O. G. & David E. E. (2014). Rank of Maximal subgroup of a full transformation semigroup. International Journal of Current Research, Vol., 6. Issue, 09, pp 8351-8354 DOI 10.5281/zenodo.10511744 IJO JOURNALS Volume 07 | Issue 01 | January 2024 | http://ijojournals.com/index.php/m/index 16 https://doi.org/10.56201/ijcsmt.v9.no5.2023.pg102.113 https://doi.org/10.5281/zenodo.10443958