Conjugacy classes of maximal cyclic subgroups of metacyclic $p$-groups

TL;DR

This paper computes the number of conjugacy classes of maximal cyclic subgroups in all metacyclic p-groups, revealing lower bounds and conditions for equality, thus advancing understanding of their subgroup structure.

Contribution

It provides a complete calculation of conjugacy classes of maximal cyclic subgroups in metacyclic p-groups and characterizes cases of minimal class counts.

Findings

ta(G) is computed for all metacyclic p-groups.

ta(G)   n-2 for most non-dihedral, quaternion, or semi-dihedral groups.

Conditions for equality ta(G) = n-2 are explicitly determined.

Abstract

In this paper, we set $\eta (G)$ to be the number of conjugacy classes of maximal cyclic subgroups of a finite group $G$. We compute $\eta (G)$ for all metacyclic $p$-groups. We show that if $G$ is a metacyclic $p$-group of order $p^n$ that is not dihedral, generalized quaternion, or semi-dihedral, then $\eta (G) \ge n-2$, and we determine when equality holds.