On characterization of groups by isomorphism type of Gruenberg-Kegel graph

Abstract

The Gruenberg-Kegel graph (or the prime graph) $\Gamma (G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of order $rs$ in $G$. A group $G$ is recognizable by isomorphism type of Gruenberg--Kegel graph if for every group $H$ the isomorphism between $\Gamma (H)$ and $\Gamma (G)$ as abstract graphs (i.\,e. unlabeled graphs) implies that $G\cong H$. In this paper, we prove that finite simple exceptional groups of Lie type ${}^2{E}_6(2)$ and $E_8(q)$ for $q\in \{3,4,5,7,8,9,17\}$ are recognizable by isomorphism type of Gruenberg-Kegel graph.

15 pages, 3 figures