On The Unrecognizability by Prime Graph for the Most Simple Group ${\rm {\bf PGL(2,9)}}$

Abstract

The prime graph of a finite group $G$ is denoted by $\ga(G)$. Also $G$ is called recognizable by prime graph if and only if each finite group $H$ with $\ga(H)=\ga(G)$, is isomorphic to $G$. In this paper, we classify all finite groups with the same prime graph as $\textrm{PGL}(2,9)$. In particular, we present some solvable groups with the same prime graph as $\textrm{PGL}(2,9)$.