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

Part of: Representation theory of groups Conditions on elements Rings and algebras arising under various constructions

Published online by Cambridge University Press:  16 March 2017

Andreas Bächle  and
Leo Margolis
Andreas Bächle
Affiliation:
Vakgroep Wiskunde, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium (abachle@vub.ac.be)
Leo Margolis
Affiliation:
Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany (leo.margolis@mathematik.uni-stuttgart.de)
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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 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 M10 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.

Keywords

integral group ringtorsion unitZassenhaus conjectureprime graph question

MSC classification

Secondary: 16U60: Units, groups of units 16S34: Group rings , Laurent polynomial rings 20C05: Group rings of finite groups and their modules 20C10: Integral representations of finite groups
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 4 , November 2017 , pp. 813 - 830
Copyright
Copyright © Edinburgh Mathematical Society 2017 

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