On characterization by Gruenberg-Kegel graph of finite simple exceptional groups of Lie type

TL;DR

This paper investigates the Gruenberg-Kegel graph of finite simple exceptional groups of Lie type, establishing conditions under which these groups are uniquely characterized by their graphs, and identifying certain groups as unrecognizable.

Contribution

It proves that most finite simple exceptional groups of Lie type with at least three connected components in their Gruenberg-Kegel graph are almost recognizable, and identifies specific groups as unrecognizable.

Findings

Most exceptional groups with ≥3 connected components are almost recognizable.

Groups ${^2}B_2(2^{2n+1})$ and $G_2(3)$ are unrecognizable.

Provides criteria linking group structure to Gruenberg-Kegel graph properties.

Abstract

The Gruenberg-Kegel 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 finite group $G$ is called almost recognizable (by Gruenberg-Kegel graph) if there is only finite number of pairwise non-isomorphic finite groups having Gruenberg-Kegel graph as $G$. If $G$ is not almost recognizable, then it is called unrecognizable (by Gruenberg--Kegel graph). Recently P.J. Cameron and the first author have proved that if a finite group is almost recognizable, then the group is almost simple. Thus, the question of which almost simple groups (in particular, finite simple groups) are almost recognizable is of prime interest. We prove that every finite simple exceptional group of Lie type, which is isomorphic to neither ${^2}B_2(2^{2n+1})$ with $n\geq1$ nor $G_2(3)$ and whose Gruenberg-Kegel graph has at least three connected components, is almost recognizable. Moreover, groups $ {^2}B_2(2^{2n+1})$, where $n\geq1$, and $G_2(3)$ are unrecognizable.