A new characterization of some alternating and symmetric groups

Abstract

We suppose that p=2α3β+1, where α≥1, β≥0, and p≥7 is a prime number. 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.