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(S4), where  S4 is the symmetric group on four elements. Moreover, we prove that G ∼= 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.