Characterizing some finite groups by the average order (Q6648818)

Let \(G\) be a finite group, \(\psi(G) =\sum_{g \in G} o(g)\) and let \(o(G)=\psi(G)/|G|\) be the average order of \(G\). As an example, \(\psi(A_{4})=\psi(D_{10})=31\) and \(o(D_{12})=o(C_{4})=\frac{11}{4}\), so such quantities are not always usable to identify a group. If \(G\) is a group such that \(o(H)=o(G)\) implies \(H \simeq G\), then \(G\) is called characterizable by average order.\N\NIn the paper under review, the authors classify groups whose average orders are less than \(o(S_{4})\) and they prove that \(S_{4}\) is characterizable by average order. \textit{M. Tărnăuceanu}, in [J. Algebra 604, 682--693 (2022; Zbl 1520.20035)], has determined the groups whose average orders is less than \(o(A_{4})\). The authors point out that they used independent proof methods which led them to regain the result cited above.