A result on the sum of element orders of a finite group (Q2285057)

In this interesting paper, the authors discuss the relationship between the sum of the element orders of a finite group and the structure of the group. Let \(\psi(G)\) denote the sum of the element orders of \(G\) and let \(C_n\) denote a cyclic group of order \(n\). Then Herzog conjectured the following: If \(n \in \mathbb{N}\) and if \(G\) is a non-soluble group of order \(n\), then \(\psi(G) \le \frac{211}{1617} \cdot \psi(C_n)\), with equality if and only if \(G \cong \mathcal{A}_5\). The authors prove a version of this conjecture, which also gives a solubility criterion for finite groups. In particular, they show that if the bound in the conjecture is attained, then the group \(G\) must be a direct product of \(\mathcal{A}_5\) with a cyclic group of order coprime to \(30\).