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

TL;DR

This paper classifies all finite groups sharing the prime graph with PGL(2,9), revealing that some solvable groups are indistinguishable from PGL(2,9) based solely on their prime graph.

Contribution

It provides a complete classification of groups with the same prime graph as PGL(2,9), including solvable groups, highlighting unrecognizability by prime graph.

Findings

Identified all groups with the same prime graph as PGL(2,9)

Discovered solvable groups sharing the prime graph with PGL(2,9)

Showed PGL(2,9) is unrecognizable by prime graph

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 $\textrm{PGL}(2,9)$.