On the prime graph of simple groups

Abstract

Let G be a finite group, let π(G) be the set of prime divisors of jGj and let Γ(G) be the prime graph of G. This graph has vertex set π(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 Γ(G) = Γ(H).