A Criterion for Solvability of a Finite Group by the Sum of Element Orders

TL;DR

This paper proves a conjecture linking the sum of element orders in a finite group to its solvability, establishing a criterion based on a specific inequality involving cyclic groups.

Contribution

It confirms a conjecture that characterizes the solvability of finite groups through a new inequality involving the sum of element orders.

Findings

The conjecture relating $ ext{sum of element orders}$ to solvability is proven.

A new solvability criterion based on element order sums is established.

The result applies to all finite groups satisfying the inequality.

Abstract

Let $G$ be a finite group and $\psi(G) = \sum_{g \in G} o(g)$, where $o(g)$ denotes the order of $g \in G$. In [M. Herzog, et. al., Two new criteria for solvability of finite groups, J. Algebra, 2018], the authors put forward the following conjecture: \textbf{Conjecture.} \textit{If $G$ is a group of order $n$ and $\psi(G)>211\psi(C_n)/1617 $, where $C_n$ is the cyclic group of order $n$, then $G$ is solvable.} In this paper we prove the validity of this conjecture.