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

Morteza Baniasad Azad; Behrooz Khosravi

paper

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

arXiv:1808.00253

Abstract

Let $G$ be a finite group and $ψ (G) = \sum_{g \in G} o (g)$ , 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 $ψ (G) > 211 ψ (C_{n}) /1617$ , where $C_{n}$ is the cyclic group of order $n$ , then $G$ is solvable.} In this paper we prove the validity of this conjecture.