Simple groups are characterized by their non-commuting graphs

Abstract

Abstract. The non-commuting graph of a finite group G is a highly symmetrical object (indeed, embeds in ), yet its complexity pales in comparison to that of G. Still, it is natural to seek conditions under which G can be reconstructed from . Surely some conditions are necessary, as is evidenced by the minuscule example . A conjecture made in [J. Algebra 298 (2006), 468–492], commonly referred to as the AAM Conjecture, proposes that the property of being a nonabelian simple group is sufficient. In [Sib. Math. J. 49 (2008), no. 6, 1138–1146], this conjecture is verified for all sporadic simple groups, while in [J. Algebra 357 (2012), 203–207], it is verified for the alternating groups. In this paper we verify it for the simple groups of Lie type, thereby completing the proof of the conjecture.