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ON THE PRIME GRAPH OF SIMPLE GROUPS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 91 / Issue 2 / April 2015
- Published online by Cambridge University Press:
- 08 October 2014, pp. 227-240
- Print publication:
- April 2015
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- Article
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- You have access
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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 
${\rm\Gamma}(G)={\rm\Gamma}(H)$.
