Criterion of unrecognizability of a finite group by its Gruenberg-Kegel graph

TL;DR

This paper investigates the properties and recognition criteria of finite groups based on their Gruenberg-Kegel graphs, providing new results on group classification, graph properties, and the uniqueness of certain groups.

Contribution

The paper introduces new criteria for recognizing finite groups from their Gruenberg-Kegel graphs and classifies groups with specific graph properties, including those uniquely determined by their graphs.

Findings

Infinite groups can share the same Gruenberg-Kegel graph under certain conditions.

A polynomial bound on the number of groups sharing a given Gruenberg-Kegel graph.

Groups with finite Gruenberg-Kegel graphs are almost simple and have specific structural properties.

Abstract

The Gruenberg-Kegel graph $\Gamma(G)$ associated with a finite group $G$ has as vertices the prime divisors of $|G|$, with an edge from $p$ to $q$ if and only if $G$ contains an element of order $pq$. This graph has been the subject of much recent interest; one of our goals here is to give a survey of some of this material, relating to groups with the same Gruenberg-Kegel graph. However, our main aim is to prove several new results. Among them are the following. - There are infinitely many finite groups with the same Gruenberg-Kegel graph as the Gruenberg-Kegel of a finite group $G$ if and only if there is a finite group $H$ with non-trivial solvable radical such that $\Gamma(G)=\Gamma(H)$. - There is a function $F$ on the natural numbers with the property that if a finite $n$-vertex graph whose vertices are labelled by pairwise distinct primes is the Gruenberg-Kegel graph of more than $F(n)$ finite groups, then it is the Gruenberg-Kegel graph of infinitely many finite groups. (The function we give satisfies $F(n)=O(n^7)$, but this is probably not best possible.) - If a finite graph $\Gamma$ whose vertices are labelled by pairwise distinct primes is the Gruenberg-Kegel graph of only finitely many finite groups, then all such groups are almost simple; moreover, $\Gamma$ has at least three pairwise non-adjacent vertices, and $2$ is non-adjacent to at least one odd vertex. - Groups whose power graphs, or commuting graphs, are isomorphic have the same Gruenberg-Kegel graph. - The groups ${^2}G_2(27)$ and $E_8(2)$ are uniquely determined by the isomorphism types of their Gruenberg-Kegel graphs. In addition, we consider groups whose Gruenberg-Kegel graph has no edges. These are the groups in which every element has prime power order, and have been studied under the name \emph{EPPO groups}; completing this line of research, we give a complete list of such groups.