On the composition factors of a group with the same prime graph as B n (5)

Abstract

Let G be a finite group. The prime graph of G is a graph whose vertex set is the set of prime divisors of |G| and two distinct primes p and q are joined by an edge, whenever G contains an element of order pq. The prime graph of G is denoted by Γ(G). It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if G is a finite group such that Γ(G) = Γ(B n (5)), where n ⩾ 6, then G has a unique nonabelian composition factor isomorphic to B n (5) or C n (5).