© Hindawi Publishing Corp. A NEW CHARACTERIZATION OF SOME ALTERNATING AND SYMMETRIC GROUPS

We suppose that p = 2α3β +1, where α ≥ 1, β ≥ 0, and p ≥ 7 is a prime num-ber. Then we prove that the simple groups An, where n = p,p+1, or p+2, and finite groups Sn, where n = p,p+1, are also uniquely determined by their order components. As corollaries of these results, the validity of a conjecture of J. G. Thompson and a conjecture of Shi and Bi (1990) both on An, where n = p,p+1, or p+2, is obtained. Also we generalize these conjectures for the groups Sn, where n = p,p+1. 2000 Mathematics Subject Classification: 20D05, 20D60, 20D08. 1. Introduction. Let G be a finite group. We denote by π(G) the set of all prime divisors of |G|. We construct the prime graph of G as follows. The prime graph Γ(G) of a groupG is the graph whose vertex set isπ(G), and two distinct primes p and q are joined by an edge (we write p ∼ q) if and only if G contains an element of order pq. Let t(G) be the number of connected components o