Another criterion for supersolvability of finite groups

TL;DR

This paper establishes a new criterion for supersolvability of finite groups based on the average order, proving that groups with average order less than 31/12 are supersolvable and classifying those with average order exactly 31/12.

Contribution

It introduces a novel average order criterion for supersolvability and characterizes the groups achieving the threshold.

Findings

Groups with average order less than 31/12 are supersolvable.

The only group with average order exactly 31/12 is A_4.

Finite groups with average order below 31/12 are classified.

Abstract

Let $o(G)$ be the average order of a finite group $G$. In this paper, we prove that if $o(G)<\frac{31}{12}$\,, then $G$ is supersolvable. Moreover, we have $o(G)=\frac{31}{12}$ if and only if $G\cong A_4$. We also classify finite groups $G$ satisfying $o(G)<\frac{31}{12}$\,.