ON THE RECOGNITION OF THE SIMPLE GROUPS L₇(3) AND L₈(3) BY THE SPECTRUM

Abstract

In this paper we will prove that up to isomorphism there are two finite groups with the same spectrum as that of the simple group L 7 (3). For the group L 8 (3) we will prove that if G is a finite group with the same spectrum as that of L 8 (3), then G is isomorphic to the group L 8 (3).

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

“…In this paper as the main result we prove that the simple group PSL n (3), where n 9, is quasirecognizable by prime graph; i.e., if G is a finite group such that Γ(G) = Γ(PSL n (3)), then G has a unique nonabelian composition factor isomorphic to PSL n (3). In [8], it is proved that the projective special linear group PSL p (3), where p > 3 is a prime number, is at most 2-recognizable by spectrum, i.e., if G is a finite group such that ω(G) = ω(PSL p (3)), where p > 3 is an odd prime, then G is isomorphic to PSL p (3) or PSL p (3) • 2, the extension of PSL p (3) by the graph automorphism. As a consequence of our result we prove that if n 9, then PSL n (3) is at most 2-recognizable by spectrum, i.e., if G is a finite group such that ω(G) = ω(PSL n (3)), then G is isomorphic to PSL n (3) or PSL n (3) • 2, the extension of PSL n (3) by the graph automorphism.…”