Rational conjugacy of torsion units in integral group rings of non-solvable groups

TL;DR

This paper introduces a new method using representation theory to study rational conjugacy of torsion units in integral group rings, verifying the Zassenhaus Conjecture for specific non-solvable groups and addressing the Prime Graph Question.

Contribution

It presents a novel approach to rational conjugacy in integral group rings and proves the Zassenhaus Conjecture for PSL(2,19) and PSL(2,23), and confirms the Prime Graph Question for certain groups.

Findings

Verified Zassenhaus Conjecture for PSL(2,19) and PSL(2,23)

Showed no units of order 6 in integral group rings of M10 and PGL(2,9)

Confirmed Prime Graph Question for groups with order divisible by at most three primes

Abstract

We introduce a new method to study rational conjugacy of torsion units in integral group rings using integral and modular representation theory. Employing this new method, we verify the first Zassenhaus Conjecture for the group $\operatorname{PSL}(2,19)$. We also prove the Zassenhaus Conjecture for $\operatorname{PSL}(2,23)$. In a second application we show that there are no normalized units of order $6$ in the integral group rings of $M_{10}$ and $\operatorname{PGL}(2,9)$. This completes the proof of a theorem of W. Kimmerle and A. Konovalov that the Prime Graph Question has an affirmative answer for all groups having an order divisible by at most three different primes.