Quasirecognition by the Prime Graph of the Group Cn(2), Where n≠3 is Odd

Let G be a finite group and let Γ(G) be the prime graph of G. We assume that n is an odd number. In this paper, we show that if Γ(G)=Γ(Cn(2)), where n≠3, then G has a unique nonabelian composition factor isomorphic to Cn(2). As consequences of our result, Cn(2) is quasirecognizable by its spectrum and by a new proof the validity of a conjecture of W. J. Shi for Cn(2) is obtained.