Characterization of‎ ‎A_۵ and PSL(۲,۷) by sum of element orders

Let G be a finite group‎. ‎We denote by \psi(G) the integer \sum_{g\in G}o(g)‎, ‎where o(g) denotes the order of g \in G‎. ‎Here we show that‎ ‎\psi(A_۵)< \psi(G) for every non-simple group G of order ۶۰‎, ‎where A_۵ is the alternating group of degree ۵‎. ‎Also we prove that \psi(PSL(۲,۷))<\psi(G) for all non-simple‎ ‎groups G of order ۱۶۸‎. ‎These two results confirm the conjecture‎ ‎posed in [J‎. ‎Algebra Appl.‎, ‎{\bf ۱۰} No‎. ‎۲ (۲۰۱۱) ۱۸۷-۱۹۰] for simple groups A_۵ and PSL(۲,۷)‎.

Keywords:

Finite groups , simple group , element orders