# Article

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Keywords:
projective general linear group; prime graph; recognition
Summary:
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.
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