Characterizing some finite groups by the 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(S ₄), where S ₄ is the symmetric group on four elements. Moreover, we prove that G ≌ S ₄ if and only if o(G) = o(S ₄). 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(A ₄) 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.