On groups having the prime graph as alternating and symmetric groups

The prime graph Γ(G) of a finite group G is the graph whose vertex set is the set of prime divisors of |G| 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 A _n (S _n ) denote the alternating (symmetric) group of degree n. We prove that if G is a finite group with Γ(G)=Γ(A _n ) or Γ(G)=Γ(S _n ), where n≥19, then there exists a normal subgroup K of G and an integer t such that A _t ≤G(K)≤S _t and |K| is divisible by at most one prime greater than n/2.