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A GROUP SUM INEQUALITY AND ITS APPLICATION TO POWER GRAPHS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 90 / Issue 3 / December 2014
- Published online by Cambridge University Press:
- 13 June 2014, pp. 418-426
- Print publication:
- December 2014
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- Article
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Let
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$ be a finite group of order 
$n$, and let 
$\text {C}_n$ be the cyclic group of order 
$n$. For 
$g\in G$, let 
${\mathrm{o}}(g)$ denote the order of 
$g$. Let 
$\phi $ denote the Euler totient function. We show that 
$\sum _{g \in \text {C}_n} \phi ({\mathrm{o}}(g))\geq \sum _{g \in G} \phi ({\mathrm{o}}(g))$, with equality if and only if 
$G$ is isomorphic to 
$\text {C}_n$. 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 bidirectional edges in its directed power graph.
