A result on the sum of element orders of a finite group

Abstract

Let G be a finite group and $$\\psi (G)=\\sum _{g\\in {G}}{o(g)}$$ ψ ( G ) = ∑ g ∈ G o ( g ) . There are some results about the relation between $$\\psi (G)$$ ψ ( G ) and the structure of G . For instance, it is proved that if G is a group of order n and $$\\psi (G)>\\dfrac{211}{1617}\\psi (C_n)$$ ψ ( G ) > 211 1617 ψ ( C n ) , then G is solvable. Herzog et al. in (J Algebra 511:215–226, 2018) put forward the following conjecture: Conjecture. If G is a non-solvable group of order n , then $$\\begin{aligned} {\\psi (G)}\\,{\\le }\\,{{\\dfrac{211}{1617}}{\\psi (C_n)}}, \\end{aligned}$$ ψ ( G ) ≤ 211 1617 ψ ( C n ) , with equality if and only if $$G \\cong A_5$$ G ≅ A 5 . In particular, this inequality holds for all non-Abelian simple groups. In this paper, we prove a modified version of Herzog’s Conjecture.