R-Recognizability of [formula omitted] and [formula omitted] where [formula omitted

Abstract

Let G be a finite group and OC ( G ) be the set of order components of G. Denote by k ( OC ( G ) ) the number of isomorphism classes of finite groups H satisfying OC ( H ) = OC ( G ) . It is proved that some finite groups are uniquely determined by their order components, i.e. k ( OC ( G ) ) = 1 . Let n = 2 m ⩾ 4 . As the main result of this paper, we prove that if q is odd, then k ( OC ( B n ( q ) ) ) = k ( OC ( C n ( q ) ) ) = 2 and if q is even, then k ( OC ( C n ( q ) ) ) = 1 . A main consequence of our results is the validity of a conjecture of J.G. Thompson and another conjecture of W. Shi and J. Bi for the groups C n ( q ) , where n = 2 m ⩾ 4 and q is even.