Characterization of $A_5$ and $PSL(2,7)$ by sum of element orders

Abstract

Let $G$ be a finite group‎. ‎We denote by $psi(G)$ the integer $sum_{gin G}o(g)$‎, ‎where $o(g)$ denotes the order of $g in G$‎. ‎Here we show that‎ ‎$psi(A_5)< psi(G)$ for every non-simple group $G$ of order $60$‎, ‎where $A_5$ is the alternating group of degree $5$‎. ‎Also we prove that $psi(PSL(2,7))<psi(G)$ for all non-simple‎ ‎groups $G$ of order $168$‎. ‎These two results confirm the conjecture‎ ‎posed in [J‎. ‎Algebra Appl.‎, ‎{bf 10} No‎. ‎2 (2011) 187-190] for simple groups $A_5$ and $PSL(2,7)$‎.