### 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 \(n\). For \(g∈G\), let \(o(g)\) denote the order of \(g\). Let \(ϕ\) denote the Euler totient function. We show that \(\sum_{g \in C_n} \phi(o(g))\geq \sum_{g \in G} \phi(o(g))\), 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.

DOI:-

10.1017/S0004972714000434

DOI:-

10.1017/S0004972714000434

#### Keywords

Cyclic groups, Euler totient, Sylow subgroups

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Bulletin of the Aust. Math. Soc., copyright Australian Mathematical Publishing Association Inc.