A characterization of \(A_5\) by its average order
Let \(G\) be a finite group, \(\psi(G)=\sum_{g \in G} o(g)\) the sum of element orders of \(G\) and \(o(G)=\psi(G)/|G|\) the average order of \(G\). In [\textit{M. Herzog} et al., J. Algebra 597, 1--23 (2022; Zbl 1487.20008)] it was proven that if \(G\) is non-solvable and \(o(G)=o(A_{5})=\frac{211}{60}\), then \(G \simeq A_{5}\).\N\NIn the paper under review, the author proves that the equality \(o(G)=o(A_{5})\) does not hold for any finite solvable group \(G\). In particular, this constitutes a new proof of the fact that \(A_{5}\) is determined by its average order.
scientific article; zbMATH DE number 7950843