On groups having the prime graph as alternating and symmetric groups

Abstract

The prime graph of a finite group G is the graph whose vertex set is the set of prime divisors of and in which two distinct vertices r and s are adjacent if and only if there exists an element of G of order rs. Let An (Sn) denote the alternating (symmetric) group of degree n. We prove that if G is a finite group with or , where , then there exists a normal subgroup K of G and an integer t such that and is divisible by at most one prime greater than .