Resumen de A New Characterization of Some Alternating and Symmetric Groups (II)

Behrooz Khosravi, Amir Khosravi

• The order of every finite group G can be expressed as a product of coprime positive integers m1,...,mt such that the set of prime numbers divided mi is a connected component of the prime graph of G. The integers m1,...,mt are called the order components of G. Order components of a finite group are introduced in Chen (J. Algebra 15 (1996) 184).

  There exist some characterizations about alternating and symmetric groups. Some non-abelian simple groups are known to be uniquely determined by their order components. In this paper, we suppose that p=2a xb+1>5 be a prime number, where a,b>0 are positive integers and x>3 is an odd prime number. Then by using the classification of finite simple groups, we proved that Ap, Ap+1, Ap+2, Sp, Sp+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 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.