Cyclic Subgroups and Conjugacy Classes in Finite p-Groups of Small Exponent

Abstract

Finite p-groups of exponent p or p² show rather rigid conjugacy behaviour between their cyclic subgroups. These restrictions give rise to invariants pertaining to nilpotency class and isomorphism type of derived subgroup. This survey unifies previously known bounds and classifications about conjugacy classes of cyclic subgroups. Two quantities are critical. The first is η(G), which counts the conjugacy classes of maximal cyclic subgroups. The second is the number of conjugacy classes of non-normal cyclic subgroups, denoted by v_c (G). For a group G of order pⁿ and nilpotency class l, it holds that: η(G) ≥ (p - 1)(n/l - 2) + p + 1. This is linear in n, if p and l are considered fixed. For other classes, like metacyclic groups and finitely-generated groups of exponent p, sharper bounds are known. For any odd primes, groups with v_c (G)= p and v_c (G),= p + 1 have been classified, fixing the lower end of the spectrum. There is also the evidence of a connection between v_c (G) and the commutator subgroup. The case with |G'| = pᵏ supports the result v_c (G) ≥ k. This connects the number of non-normal cyclic subgroups and size of derived subgroup. Elementary calculations for groups of order at most p⁶, for the most part with p = 2 and also sometimes with p = 3 confirm these patterns. The data indicates that η(G) increases linearly with n. The lower bounds were equal or almost in certain families. These include metacyclic, modular, dihedral and generalised quaternion groups and some class two ones. Low values of v_c (G) occur when many cyclic subgroups are normal and the central quotient is small. This does not require many steps to solve because conjugation is limited to a very small range of cyclic subgroups, due to small exponent, and the normalizer condition reduces the number of those failing to be normal. These results give a scaling behavior for v_c (G) over finite p-groups and allow to separate structural families more finely. Keywords: Finite p groups, Conjugacy Classes, Maximal Cyclic Subgroups, Nilpotency Class, Patterns, Subgroups, Non-Normal Subgroups. Amao, F.A., Oladapo, D.I., Akinsola, V. O. & Ilo, U. C. (2026): Cyclic Subgroups and Conjugacy Classes in Finite p-Groups of Small Exponent. Journal of Advances in Mathematical & Computational Science. Vol. 14, No. 1. Pp 57-66. Available online at www.isteams.net/mathematics-computationaljournal. dx.doi.org/10.22624/AIMS/MATHS/V14N1P4