On some Frobenius groups with the same prime graph as the almost simple group ${ {\bf PGL(2,49)}}$

Subject Areas : Group theory

A. Mahmoudifar ¹

1 - Department of Mathematics, Tehran North Branch, Islamic Azad University, Tehran, Iran

Received: 2016-12-25 Accepted : 2017-04-15 Published : 2017-10-01

Keywords: prime graph, Almost simple group, element order, Frobenius group,

Abstract :

The prime graph of a finite group $G$ is denoted by$\Gamma(G)$ whose vertex set is $\pi(G)$ and two distinct primes $p$ and $q$ are adjacent in $\Gamma(G)$, 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 $\Gamma(H)=\Gamma(G)$, 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 groupswhose prime graphs coincide with $\Gamma(\textrm{PGL}(2,49))$, in particular, we get that $\textrm{PGL}(2,49)$ is unrecognizable by its prime graph.

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