On some Frobenius groups with the same prime graph as the almost simple group PGL(2,49)

Abstract

The prime graph of a finite group $G$ is denoted by $\ga(G)$ whose vertex set is $π(G)$ and two distinct primes $p$ and $q$ are adjacent in $\ga(G)$, whenever $G$ contains an element with order $pq$. We say that $G$ is unrecognizable by prime graph if there is a finite group $H$ with $\ga(H)=\ga(G)$, in while $H\not\cong G$. In this paper, we consider finite groups with the same prime graph as the almost simple group $\textrm{PGL}(2,49)$. Moreover, we construct some Frobenius groups whose their prime graph coincide with $\ga(\textrm{PGL}(2,49))$, in particular, we get that $\textrm{PGL}(2,49)$ is unrecognizable by prime graph.