r-Recognizability of Bn(q) and Cn(q) where n=2m⩾4

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=2m⩾4. As the main result of this paper, we prove that if q is odd, then k(OC(Bn(q)))=k(OC(Cn(q)))=2 and if q is even, then k(OC(Cn(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 Cn(q), where n=2m⩾4 and q is even.