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ON THE PRIME GRAPH OF SIMPLE GROUPS

Published online by Cambridge University Press:  08 October 2014

TIMOTHY C. BURNESS*
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK email t.burness@bristol.ac.uk
ELISA COVATO
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK email elisa.covato@bristol.ac.uk
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Abstract

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Let $G$ be a finite group, let ${\it\pi}(G)$ be the set of prime divisors of $|G|$ and let ${\rm\Gamma}(G)$ be the prime graph of $G$. This graph has vertex set ${\it\pi}(G)$, and two vertices $r$ and $s$ are adjacent if and only if $G$ contains an element of order $rs$. Many properties of these graphs have been studied in recent years, with a particular focus on the prime graphs of finite simple groups. In this note, we determine the pairs $(G,H)$, where $G$ is simple and $H$ is a proper subgroup of $G$ such that ${\rm\Gamma}(G)={\rm\Gamma}(H)$.

MSC classification

Type
Research Article
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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