A result on the sum of element orders of a finite group

Abstract

Let $G$ be a finite group and $\psi (G)={\sum }_{g\in G}o(g)$. There are some results about the relation between $\psi (G)$ and the structure of $G$. For instance, it is proved that if $G$ is a group of order $n$ and $\psi (G)>\frac{211}{1617}\psi (C_n)$, then $G$ is solvable. Herzog {\it{et al.}} in [Herzog {\it{et al.}}, Two new criteria for solvability of finite groups, J. Algebra, 2018] put forward the following conjecture: \noindent{\bf Conjecture.} {\it {If $G$ is a non-solvable group of order $n$, then $$\psi (G)\le \frac{211}{1617}\psi (C_n)$$ with equality if and only if $G=A_5$. In particular, this inequality holds for all non-abelian simple groups.} } In this paper, we prove a modified version of Herzog's Conjecture.

9 pages