NONCOMMUTING GRAPH CHARACTERIZATION OF SOME SIMPLE GROUPS WITH CONNECTED PRIME GRAPHS

Authors : Liangcai ZHANG, Wujie SHİ

Pages : 169-181

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Publication Date : 2009-06-01

Article Type : Research Paper

Abstract :Let G be a nonabelian group and associate a noncommuting graph ∇insert ignore into journalissuearticles values(G); with G as follows: The vertex set of ∇insert ignore into journalissuearticles values(G); is G\Zinsert ignore into journalissuearticles values(G);, where Zinsert ignore into journalissuearticles values(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 ∇insert ignore into journalissuearticles values(G); ∼= ∇insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(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 L4insert ignore into journalissuearticles values(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 L4insert ignore into journalissuearticles values(4);, L4insert ignore into journalissuearticles values(7);, U4insert ignore into journalissuearticles values(7);, etc.
Keywords : finite group, noncommuting graph, projective special linear simple group