A group sum inequality and its application to power graphs

Abstract

Let $G$ be a finite group of order $n$, and let $C_n$ be the cyclic group of order $n$. We show that ${\sum }_{g\in C_n}\varphi (\mathrm{o}(g))\ge {\sum }_{g\in G}\varphi (\mathrm{o}(g))$, with equality if and only if $G$ is isomorphic to $C_n$. As an application, we show that among all finite groups of a given order, the cyclic group of that order has the maximum number of undirected edges in its directed power graph.