Characterizing some finite groups by the average order

TL;DR

This paper classifies finite groups based on their average order, specifically identifying those with average order less than that of S4 and characterizing S4 uniquely by its average order, providing new insights without relying on previous main theorems.

Contribution

It introduces a novel classification of finite groups by average order and characterizes S4 uniquely through this property, offering alternative proofs to existing results.

Findings

Groups with average order less than o(S4) are classified.

G is isomorphic to S4 if and only if o(G) equals o(S4).

The results reprove and extend previous classifications based on average order.

Abstract

The average order of a finite group G is denoted by o(G). In this note, we classify groups whose average orders are less than o(S4), where S4 is the symmetric group on four elements. Moreover, we prove that G \cong S4 if and only if o(G) = o(S4). As a consequence of our results we give a characterization for some finite groups by the average order. In [9, Theorem 1.2], the groups whose average orders are less than o(A4) are classified. It is worth mentioning that to get our results we avoid using the main theorems of [9] and our results leads to reprove those theorems.