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

TL;DR

This paper investigates the prime graph of the group PGL(2,49), constructs Frobenius groups with the same prime graph, and demonstrates that PGL(2,49) is unrecognizable solely by its prime graph.

Contribution

It constructs specific Frobenius groups sharing the prime graph with PGL(2,49) and proves PGL(2,49) is unrecognizable by prime graph.

Findings

Constructed Frobenius groups with the same prime graph as PGL(2,49)

Proved PGL(2,49) is unrecognizable by prime graph

Enhanced understanding of prime graph recognition limitations

Abstract

The prime graph of a finite group $G$ is denoted by $\ga(G)$ whose vertex set is $\pi(G)$ 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.