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

## TL;DR

This paper investigates the sum of element orders in finite groups, proving a modified version of Herzog's conjecture that relates this sum to the group's structure and solvability.

## Contribution

It provides a proof for a modified Herzog's conjecture, establishing a new inequality involving the sum of element orders and group solvability.

## Key findings

- Proves a modified version of Herzog's conjecture.
- Establishes a bound on the sum of element orders for non-solvable groups.
- Identifies the equality case as the group A_5.

## 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 $\psi(G)>\dfrac{211}{1617}\psi(C_n)$, 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 $${\psi(G)}\,{\leq}\,{{\dfrac{211}{1617}}{\psi(C_n)}}$$ 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.