A new characterization of PGL(2,p) by its noncommuting graph

Abstract

Let G be a finite non-abelian group. The noncommuting graph of G is denoted by ∇(G) and is defined as follows: the vertex set of ∇ (G) is G\Z(G) and two vertices x and y are adjacent if and only if xy ≠ yx. Let p be a prime number. In this paper, it is proved that the almost simple group PGL(2, p) is uniquely determined by its noncommuting graph. As a consequence of our results the validity of a conjecture of Thompson and another conjecture of Shi and Bi for the group PGL(2,p) are proved.