On recognition by prime graph of the projective special linear group over GF(3)

Abstract

Let G be a finite group. The prime graph of G is denoted by Γ(G). We prove that the simple group PSLn(3), where n 9, is quasirecognizable by prime graph; i.e., if G is a finite group such that Γ(G) = Γ(PSLn(3)), then G has a unique nonabelian composition factor isomorphic to PSLn(3). Darafsheh proved in 2010 that if p > 3 is a prime number, then the projective special linear group PSLp(3) is at most 2-recognizable by spectrum. As a consequence of our result we prove that if n 9, then PSLn(3) is at most 2-recognizable by spectrum.