Characterization of groups $E_6(3)$ and ${}^2{E}_6(3)$ by Gruenberg--Kegel graph

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 $L\cong {}^2{E}_6(3)$. We prove that if $G$ is a finite group such that $\Gamma (G)=\Gamma (L)$, then $G\cong L$.

Corrected graph pictures. 6 pages