On the Gruenberg-Kegel Graph of Integral Group Rings of Finite Groups

TL;DR

This paper investigates the prime graph question for integral group rings, reducing the problem to almost simple groups and confirming the graph coincidence for groups with orders divisible by up to three primes.

Contribution

It reduces the prime graph question to almost simple groups and verifies the graph coincidence for finite groups with orders divisible by at most three primes.

Findings

Reduction of the prime graph question to almost simple groups.

Confirmation of the graph coincidence for groups with order divisible by up to three primes.

Abstract

The prime graph question asks whether the Gruenberg-Kegel graph of an integral group ring $\mathbb Z G$ , i.e. the prime graph of the normalised unit group of $\mathbb Z G$ coincides with that one of the group $G$. In this note we prove for finite groups $G$ a reduction of the prime graph question to almost simple groups. We apply this reduction to finite groups $G$ whose order is divisible by at most three primes and show that the Gruenberg - Kegel graph of such groups coincides with the prime graph of $G$.