On the recognizability of some projective general linear groups by the prime graph

Let G be a finite group. The prime graph of G is a simple graph Gamma(G) whose vertex set is pi (G) and two distinct vertices p and q are joined by an edge if and only if G has an element of order pq. A group G is called k -recognizable by prime graph if there exist exactly k nonisomorphic groups H satisfying the condition Gamma(G) = Gamma(H). A 1-recognizable group is usually called a recognizable group. In this problem, it was proved that PGL(2, p alpha) is recognizable, if p is an odd prime and alpha > 1 is odd. But for even alpha, only the recognizability of the groups PGL(2, 52), PGL(2, 32) and PGL(2, 34) was investigated. In this paper, we put alpha = 2 and we classify the finite groups G that have the same prime graph as Gamma(PGL(2,p2)) for p = 7,11,13 and 17. As a result, we show that PGL(2, 72) is unrecognizable; and PGL(2, 132) and PGL(2, 172) are recognizable by prime graph.