On the sum of element orders of PSL(2, p) for some p

Abstract

Let G be a finite group and ψ(G) = \(\sum_{g∈G} o(g)\), where o(g) denotes the order of g ∈ G.  In this paper, we show that if G is a group of order |PSL(2,p)|, then ψ(G) ≥ ψ(PSL(2,p)), where p ∈ {11,13,19,23,29,37,61}.  Also we prove that |G| = |PSL(2,p)| and ψ(G) = ψ(PSL(2,p)) if and only if G \(\cong\) PSL(2,p), where p ∈ {11,13,19,23,29,37,61}.  Furthermore, we prove that PSL(2,17) is determined by its order and the sum of element orders.