Bulletin

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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.

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