Recognition of some simple groups by their noncommuting graphs

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) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. Abdollahi et al. (J Algebra 298(2):468–492, 2006) put forward a conjecture called AAM’s Conjecture in 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 well known to hold for all simple groups with nonconnected prime graphs and the alternating group A10 [see Darafsheh (Groups with the same non-commuting graph. Discrete Appl Math (2008) doi:10.1016/j.dam.2008.06.010), Wang and Shi (Commun Algebra 36(2):523–528, 2008)], it is still unknown for all simple groups with connected prime graphs except A10. In the present paper, we prove that this conjecture is also true for the projective special linear simple group L4(9). The new method used in this paper also works well in the cases L4(4), L4(7), U4(7), etc.