On the Prime Graph Question for Integral Group Rings of 4-primary groups I

TL;DR

This paper investigates the Prime Graph Question for integral group rings, proving it for certain almost simple groups and analyzing the effectiveness of the HeLP-method for groups with limited prime divisors.

Contribution

It establishes the Prime Graph Question for all almost simple groups with socle isomorphic to PSL(2, p^f) for f ≤ 2, and assesses the HeLP-method's reach for groups with up to four prime divisors.

Findings

Prime Graph Question confirmed for specific almost simple groups.

HeLP-method's limitations quantified for groups with ≤4 prime divisors.

First proof of Prime Graph Question for automorphic extensions of certain simple groups.

Abstract

We study the Prime Graph Question for integral group rings. This question can be reduced to almost simple groups by a result of Kimmerle and Konovalov. We prove that the Prime Graph Question has an affirmative answer for all almost simple groups having a socle isomorphic to $\operatorname{PSL}(2, p^f)$ for $f \leq 2$, establishing the Prime Graph Question for the first time for all automorphic extensions of series of simple groups. Using this, we determine exactly how far the so-called HeLP-method can take us for (almost simple) groups having an order divisible by at most $4$ different primes.