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

TL;DR

This work studies whether finite simple groups are identifiable from the isomorphism type of their Gruenberg–Kegel (prime) graph. It proves that the exceptional groups ${}^2E_6(2)$ and $E_8(q)$ for $q\in\{3,4,5,7,8,9,17\}$ are recognizably determined by their unlabeled GK graphs: if $\Gamma(G) \cong \Gamma(H)$, then $G\cong H$ for these groups. The authors refine previous results, providing a revised proof for $E_8(q)$ and combining GK-graph analysis with table-based exclusions of all other almost-simple candidates to establish the characterizations. The findings advance the program of characterizing finite groups via prime graphs and highlight the feasibility of isomorphism-type recognizability for certain large Lie-type groups.

Abstract

The Gruenberg-Kegel graph (or the prime graph) $Γ(G)$ 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 $Γ(H)$ and $Γ(G)$ 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.

Theorems & Definitions (20)

• Lemma 1.1: Bang–Zsigmondy
• Lemma 1.2: Herzog
• Lemma 1.3
• proof
• Lemma 1.4: Gruenberg--Kegel Theorem, follows from Williams and Thompson
• Lemma 1.5: ak
• Lemma 1.6
• proof
• Lemma 1.7: Va05
• Lemma 1.8