Characterization of groups $E_6(3)$ and ${^2}E_6(3)$ by Gruenberg--Kegel graph

TL;DR

This paper proves that the Gruenberg--Kegel graph uniquely characterizes the groups $E_6(3)$ and ${^2}E_6(3)$, meaning any group with the same prime graph is isomorphic to these groups.

Contribution

It establishes the uniqueness of the Gruenberg--Kegel graph for the groups $E_6(3)$ and ${^2}E_6(3)$ among finite groups.

Findings

The prime graph of $E_6(3)$ is unique to it.

The prime graph of ${^2}E_6(3)$ is unique to it.

Any finite group with the same prime graph is isomorphic to these groups.

Abstract

The Gruenberg--Kegel graph (or the prime graph) $\Gamma(G)$ of a finite group $G$ is defined as follows. The vertex set of $\Gamma(G)$ is the set of all prime divisors of the order of $G$. Two distinct primes $r$ and $s$ regarded as vertices are adjacent in $\Gamma(G)$ if and only if there exists an element of order $rs$ in $G$. Suppose that $L\cong E_6(3)$ or $L\cong{}^2E_6(3)$. We prove that if $G$ is a finite group such that $\Gamma(G)=\Gamma(L)$, then $G\cong L$.