International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 45, Pages 2863-2872
doi:10.1155/S0161171203202386

A new characterization of some alternating and symmetric groups.

Amir Khosravi1 and Behrooz Khosravi2

1Faculty of Mathematical Sciences and Computer Engineering, University for Teacher Education, 599 Taleghani Ave., Tehran 15614, Iran
 2241 Golnaz Street, Velenjak, Tehran 19847, Iran

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.