Conjugacy classes of maximal cyclic subgroups and nilpotence class of $p$-groups

TL;DR

This paper investigates the relationship between the number of conjugacy classes of maximal cyclic subgroups and the nilpotence class in p-groups, establishing a linear lower bound based on group order and nilpotence class.

Contribution

It introduces a bound linking conjugacy classes of maximal cyclic subgroups to the nilpotence class in p-groups, providing new insights into their structural properties.

Findings

$ ext{η}(G)$ is bounded below by a linear function in $n/l$

The bound relates conjugacy classes to group order and nilpotence class

Provides a quantitative measure connecting subgroup conjugacy and nilpotence

Abstract

In this paper, we set $\eta (G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We prove that if $G$ is a $p$-group of order $p^n$ and nilpotence class $l$, then $\eta (G)$ is bounded below by a linear function in $n/l$.