Rational Conjugacy of Torsion Units in Integral Group Rings of Non-Solvable Groups

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 PSL(2, 19). We also prove the Zassenhaus conjecture for PSL(2, 23). In a...

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Bibliographic Details
Published in:Proceedings of the Edinburgh Mathematical Society 2017-11, Vol.60 (4), p.813-830
Main Authors: Bächle, Andreas, Margolis, Leo
Format: Article
Language:English
Subjects:
Integrals
Rings (mathematics)
Torsion
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Summary: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 PSL(2, 19). We also prove the Zassenhaus conjecture for 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 PGL(2, 9). This completes the proof of a theorem of Kimmerle and Konovalov that shows that the prime graph question has an affirmative answer for all groups having an order divisible by at most three different primes.
ISSN:0013-0915
1464-3839
DOI:10.1017/S0013091516000535