Gruenberg-Kegel graphs: cut groups, rational groups and the Prime Graph Question

TL;DR

This paper classifies the Gruenberg-Kegel graphs of finite solvable cut and rational groups, explores their properties, and addresses the Prime Graph Question for these classes and related groups.

Contribution

It provides a complete classification of Gruenberg-Kegel graphs for certain finite solvable groups and verifies the Prime Graph Question for these groups.

Findings

Classified Gruenberg-Kegel graphs of finite solvable cut groups with up to three primes.

Realized most Gruenberg-Kegel graphs of finite solvable cut and rational groups.

Confirmed the Prime Graph Question for finite rational groups and most cut groups.

Abstract

The Gruenberg-Kegel graph of a group is the undirected graph whose vertices are those primes which occur as the order of an element of the group, and distinct vertices $p$, $q$ are joined by an edge whenever the group has an element of order $pq$. It reflects interesting properties of the group. A group is said to be cut if the central units of its integral group ring are trivial. This is a rich family of groups, which contains the well studied class of rational groups, and has received attention recently. In the first part of this paper we give a complete classification of the Gruenberg-Kegel graphs of finite solvable cut groups which have at most three elements in their prime spectrum. For the remaining cases of finite solvable cut groups, we strongly restrict the list of the possible Gruenberg-Kegel graphs and realize most of them by finite solvable cut groups. Likewise, we give a list of the possible Gruenberg-Kegel graphs of finite solvable rational groups and realize as such all but one of them. As an application, we completely classify the Gruenberg-Kegel graphs of metacyclic, metabelian, supersolvable, metanilpotent and $2$-Frobenius groups for the classes of cut groups and rational groups, respectively. The Prime Graph Question asks whether the Gruenberg-Kegel graph of a group coincides with that of the group of normalized units of its integral group ring. The recent appearance of a counter-example for the First Zassenhaus Conjecture on the torsion units of integral group rings has highlighted the relevance of this question. We answer the Prime Graph Question for integral group rings for finite rational groups and most finite cut groups.