The minimum sum of element orders of finite groups

Abstract

Let G be a finite group and ψ(G) = ∑ g∈G o(g), where o(g) denotes the order of g ∈ G. We show that the Conjecture 4.6.5 posed in [Group Theory and Computation, (2018) 59-90], is incorrect. In fact, we find a pair of finite groups G and S of the same order such that ψ(G) < ψ(S), with G solvable and S simple.