Volume 34 • Number 3 • 2011   • A New Characterization of $PGL(2,p)$ by its Noncommuting Graph B. Khosravi and M. Khatami Abstract. Let $G$ be a finite non-abelian group. The noncommuting graph of $G$ is denoted by $\nabla(G)$ and is defined as follows: the vertex set of $\nabla(G)$ is $G\setminus Z(G)$ and two vertices $x$ and $y$ are adjacent if and only if $xy\neq yx$. 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. 2010 Mathematics Subject Classification: 20D05, 20D60. Full text: PDF