The average element order and the number of conjugacy classes of finite groups

TL;DR

This paper investigates the average element order in finite groups, demonstrating that no polynomial lower bound exists in relation to normal subgroups' average orders, even in specific group classes, answering a previously open question.

Contribution

It provides a negative result showing the absence of polynomial bounds for average element order in finite groups, including prime-power order and abelian normal subgroups.

Findings

No polynomial lower bound for $o(G)$ in terms of $o(N)$ exists.

Counterexamples even for prime-power order groups with abelian normal subgroups.

Answers a question posed by A. Jaikin-Zapirain.

Abstract

Let $o(G)$ be the average order of the elements of $G$, where $G$ is a finite group. We show that there is no polynomial lower bound for $o(G)$ in terms of $o(N)$, where $N\trianglelefteq G$, even when $G$ is a prime-power order group and $N$ is abelian. This gives a negative answer to a question of A.~Jaikin-Zapirain.