On the average order of a finite group

TL;DR

This paper investigates the average order of finite groups, establishing bounds that classify groups as elementary abelian 2-groups or solvable, and explores the distribution and properties of these average orders.

Contribution

It provides new bounds on the average order that classify finite groups and analyzes the density and integer values of the average order set.

Findings

If o(G)<13/6, then G is elementary abelian 2-group.

If o(G)<11/4, then G is solvable.

The set of average orders is not dense in certain intervals.

Abstract

Let $o(G)$ be the average order of a finite group $G$. We show that if $o(G)<c$, where $c\in \lbrace \frac{13}{6}, \frac{11}{4}\rbrace$, then $G$ is an elementary abelian 2-group or a solvable group, respectively. Also, we prove that the set containing the average orders of all finite groups is not dense in $[a, \infty)$, for all $a\in [0, \frac{13}{6}]$. We also outline some results related to the integer values of the average order. Since group element orders is a popular research topic, we pose some open problems concerning the average order of a finite group throughout the paper.