Conjugacy classes of maximal cyclic subgroups

TL;DR

This paper investigates the structure of conjugacy classes of maximal cyclic subgroups in groups, characterizing certain normal subgroups and describing the quotient groups based on the properties of non-maximal cyclic elements.

Contribution

It introduces the invariant η(G) for counting conjugacy classes of maximal cyclic subgroups and characterizes normal subgroups that preserve η(G) under quotienting.

Findings

If G^- generates a proper subgroup, then G/⟨G^-⟩ is either an elementary abelian p-group, a Frobenius group with specific properties, or isomorphic to A_5.

The paper provides a classification of groups based on the structure of non-maximal cyclic elements.

It offers criteria for when η(G/N) equals η(G) in terms of normal subgroups N.

Abstract

In this paper, we set $\eta (G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We consider $\eta$ and direct and semi-direct products. We characterize the normal subgroups $N$ so that $\eta (G/N) = \eta (G)$. We set $G^- = \{ g \in G \mid \langle g \rangle {\rm ~is~not ~maximal~cyclic} \}$. We show if $\langle G^- \rangle < G$, then $G/\langle G^- \rangle$ is either (1) an elementary abelian $p$-group for some prime $p$, (2) a Frobenius group whose Frobenius kernel is a $p$-group of exponent $p$ and a Frobenius complement has order $q$ for distinct primes $p$ and $q$, or (3) isomorphic to $A_5$.