Characterizations of Some Alternating and Symmetric Groups
We suppose that p = 2α5β+1, where α ≥ 1,β ≥ 0, and p ≥ 11 is a prime number. Then we prove that the simple group An, where n = p,p+1, or p+2 and the finite group Sn, where n = p,p+1, are also uniquely determined by their order components. As corollaries, the validity of a conjecture of J. G. Thompson, a conjecture of AAM (ref. ) and a conjecture of Shi(ref. ) both on An, where n = p,p + 1, or p + 2 are obtained. Also we generalized these conjectures for the groups Sn, where n = p,p + 1.
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