A new characterization of some alternating and symmetric groups

Abstract. There exist some characterizations about alternating and sym-metric groups. Some non-abelian simple groups are known to be uniquely determined by their order components. In our last paper [22], we supposed that p ∗ = 2α3β + 1, where α ≥ 1, β ≥ 0, and p ∗ ≥ 7 is a prime number. Then by using the classification of finite simple Cpp groups, we proved that Ap ∗ , Ap∗+1, Ap∗+2, Sp ∗ , Sp∗+1 are also uniquely determined by their order components. In this paper we prove the same theorems for p ∗ = 2αxβ + 1, where p ∗ and x> 3 are odd prime numbers and α, β ≥ 1. As corollaries of these results, the validity of a conjecture of J. G. Thompson and a conjecture of W. Shi and J. Bi both on An, where n = p∗, p∗+1 or p∗+2 are obtained. Also we generalize these conjectures for the groups Sn, where n = p∗, p∗+1. 1