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

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 $\mathrm{PSL}(2,p^f)$ for $f\le 2$, 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.

[v4] 32 pages. Corrected statement of Lemma 3.1 (applications not affected) and filled minor gap in proof of Proposition 3.4

References in corpus (1)

• On the Prime Graph Question for Almost Simple Groups with an Alternating Socle

Cited by in corpus (3)

• On the Prime Graph Question for Integral Group Rings of 4-primary groups II
• An application of blocks to torsion units in group rings
• Algorithmic aspects of units in group rings