On the prime graph of simple groups

Abstract

Let $G$ be a finite group, let $\pi (G)$ be the set of prime divisors of $|G|$ and let $\Gamma (G)$ be the prime graph of $G$. This graph has vertex set $\pi (G)$, 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 $\Gamma (G)=\Gamma (H)$.

11 pages; to appear in Bull. Aust. Math. Soc

Cited by in corpus (1)

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