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ON THE PRIME GRAPH OF SIMPLE GROUPS

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  08 October 2014

TIMOTHY C. BURNESS  and
ELISA COVATO
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
*
✉t.burness@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 ${\rm\Gamma}(G)={\rm\Gamma}(H)$.

Keywords

finite simple groupsprime graphsmaximal subgroups

MSC classification

Primary: 20E32: Simple groups
Secondary: 20E28: Maximal subgroups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 2 , April 2015 , pp. 227 - 240
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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