NONCOMMUTING GRAPH CHARACTERIZATION OF SOME SIMPLE GROUPS WITH CONNECTED PRIME GRAPHS

Year 2009, Volume: 5 Issue: 5, 169 - 181, 01.06.2009

Liangcai Zhang Wujie Shi

Abstract

Let G be a nonabelian group and associate a noncommuting graph ∇(G) with G as follows: The vertex set of ∇(G) is G\Z(G), where Z(G) is the center of G, and two vertices are adjacent by an edge whenever they do not commute. In 2006, A. Abdollahi, S. Akbari and H. R. Maimani put forward a conjecture called AAM’s Conjecture in [1] as follows: If M is a finite nonabelian simple group and G is a group such that ∇(G) ∼= ∇(M), then G ∼= M. Even though this conjecture is known to hold for all simple groups with nonconnected prime graphs and the alternating group A10 (see [11]), it is still unknown for all simple groups with connected prime graphs except A10. In the present paper, we prove that the conjecture is also true for L4(8), the projective special linear group of degree 4 over the finite field of order 8. The new method used in this paper also works well in the case L4(4), L4(7), U4(7), etc.

Keywords

finite group, noncommuting graph, projective special linear simple group