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

Abstract

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 ${\rm 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 $ {\rm PGL}(2, 5^2)$, $ {\rm PGL}(2, 3^2) $ and $ {\rm PGL}(2, 3^4) $ was investigated. In this paper, we put $ \alpha = 2$ and we classify the finite groups $G$ that have the same prime graph as $\Gamma({\rm PGL}(2, p^2))$ for $p=7, 11, 13$ and 17. As a result, we show that ${\rm PGL}(2, 7^2)$ is unrecognizable; and ${\rm PGL}(2, 13^2)$ and ${\rm PGL}(2, 17^2)$ are recognizable by prime graph.