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

Masoumeh Sajjadi — 2022

Commentationes Mathematicae Universitatis Carolinae

Let $G$ be a finite group. The prime graph of $G$ is a simple graph $Γ (G)$ whose vertex set is $π (G)$ and two distinct vertices $p$ and $q$ are joined by an edge if and only if $G$ has an element of order $p q$ . A group $G$ is called $k$ -recognizable by prime graph if there exist exactly $k$ nonisomorphic groups $H$ satisfying the condition $Γ (G) = Γ (H)$ . A 1-recognizable group is usually called a recognizable group. In this problem, it was proved that $PGL (2, p^{α})$ is recognizable, if $p$ is an odd prime and $α > 1$ is odd. But for even $α$ , only the recognizability...

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