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

Research paper by Andreas Bächle, Leo Margolis

Indexed on: 21 Jan '16Published on: 21 Jan '16Published in: Mathematics - Representation Theory

#### Abstract

We study the Prime Graph Question for the integral group ring of all almost simple groups which have an order divisible by exactly $4$ different primes using the so-called HeLP-method and show precisely how much information this method provides for this class of groups. We also 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. This paper will be followed by a second paper in which we will apply other methods for many of the groups studied here for which the HeLP-method can not answer the Prime Graph Question.