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A GROUP SUM INEQUALITY AND ITS APPLICATION TO POWER GRAPHS

Part of: Graph theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  13 June 2014

BRIAN CURTIN  and
G. R. POURGHOLI
BRIAN CURTIN*
Affiliation:
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA email bcurtin@usf.edu
G. R. POURGHOLI
Affiliation:
School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran 14155-6455, Iran email pourgholi@ut.ac.ir
*
✉bcurtin@usf.edu
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Abstract

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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.

Keywords

cyclic groupsEuler totientSylow subgroups

MSC classification

Secondary: 20F99: None of the above, but in this section 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 3 , December 2014 , pp. 418 - 426
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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