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

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 $\mathrm{PSL}(2,19)$. We also prove the Zassenhaus Conjecture for $\mathrm{PSL}(2,23)$. In a second application we show that there are no normalized units of order $6$ in the integral group rings of $M_{10}$ and $\mathrm{PGL}(2,9)$. 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.

[v3]: 22 pages, extended outline of the new method, typos corrected

References in corpus (1)

• Zassenhaus Conjecture for Integral Group Rings of Simple Linear Groups

Cited by in corpus (7)

• On the Prime Graph Question for Integral Group Rings of 4-primary groups II
• On the Prime Graph Question for Integral Group Rings of 4-primary groups I
• On the torsion units of the integral group ring of finite projective special linear groups
• Algorithmic aspects of units in group rings
• An application of blocks to torsion units in group rings
• On the Prime Graph Question for Almost Simple Groups with an Alternating Socle
• On the first Zassenhaus conjecture and direct products