On the validity of Thompson’s conjecture for alternating groups

Let G be a group. Let π(G) be the set of prime divisor of |G|. Let GK(G) denote the graph with vertex set π(G) such that two primes p and q in π(G) are joined by an edge if G has an element of order p · q. We use s(G) to denote the number of connected components of the prime graph GK(G). Let N(G) be the set of nonidentity orders of conjugacy classes of elements in G. Some authors have proved that the groups An where n = p, p + 1, p + 2 with s(G) ≥ 2, are characterized by N(G). Then if s(G) = 1, we know that Liu and Yang proved that alternating groups Ap+3 are characterized by N(G). As the development of this topics, we will prove that if G is a finite group with trivial center and N(G) = N(Ap+4) with p+ i composite and 1 ≤ i ≤ 4, then G is isomorphic to Ap+4. Key–Words: Element order, Alternating group, Thompson’s conjecture, Conjugacy classes, Simple group.