A result on certain sums of element orders in finite groups

Abstract

Given a finite group $G$ of order $p^{n}m$, where $p$ is a prime and $p\nmid m$, we denote by ${\psi }_p(G)$ the sum of orders of $p$-parts of elements in $G$. In the current note, we prove that ${\psi }_p(G)\le {\psi }_p(C_{p^{n}m})$, where $C_{p^{n}m}$ is the cyclic group of order $p^{n}m$, and the equality holds if and only if $G$ is $p$-nilpotent of a particular type. A generalization of this result is also presented.

Accepted for publication in Results Math