Simple groups are characterized by their non-commuting graphs

Abstract

The non-commuting graph δ(G) of a finite group G is a highly symmetrical object (indeed, Aut(G) embeds in Aut(δ(G))), yet its complexity pales in comparison to that of G. Still, it is natural to seek conditions under which G can be reconstructed from δ(G). Surely some conditions are necessary, as is evidenced by the minuscule example δ(D8) ≈ δ(Q8). A conjecture made in [1], commonly referred to as the AAM Conjecture, proposes that the property of being a nonabelian simple group is sufficient. In [14], this conjecture is verified for all sporadic simple groups, while in [2], 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. © de Gruyter 2013.