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On the First Zassenhaus Conjecture and Direct Products

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

Published online by Cambridge University Press:  15 October 2018

Andreas Bächle ,
Wolfgang Kimmerle  and
Mariano Serrano
Andreas Bächle
Affiliation:
Vakgroep Wiskunde, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium Email: abachle@vub.ac.be
Wolfgang Kimmerle
Affiliation:
Fachbereich Mathematik, IGT, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany Email: kimmerle@mathematik.uni-stuttgart.de
Mariano Serrano
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, 30100, Murcia, Spain Email: mariano.serrano@um.es
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Abstract

In this paper we study the behavior of the first Zassenhaus conjecture (ZC1) under direct products, as well as the General Bovdi Problem (Gen-BP), which turns out to be a slightly weaker variant of (ZC1). Among other things, we prove that (Gen-BP) holds for Sylow tower groups, and so in particular for the class of supersolvable groups.

(ZC1) is established for a direct product of Sylow-by-abelian groups provided the normal Sylow subgroups form together a Hall subgroup. We also show (ZC1) for certain direct products with one of the factors a Frobenius group.

We extend the classical HeLP method to group rings with coefficients from any ring of algebraic integers. This is used to study (ZC1) for the direct product $G\times A$, where $A$ is a finite abelian group and $G$ has order at most 95. For most of these groups we show that (ZC1) is valid and for all of them that (Gen-BP) holds. Moreover, we also prove that (Gen-BP) holds for the direct product of a Frobenius group with any finite abelian group.

Keywords

integral group ringtorsion unitZassenhaus Conjecturedirect productFrobenius groupHeLP method

MSC classification

Primary: 16S34: Group rings , Laurent polynomial rings
Secondary: 16U60: Units, groups of units 20C05: Group rings of finite groups and their modules
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 3 , June 2020 , pp. 602 - 624
Copyright
© Canadian Mathematical Society 2018

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Footnotes

The first author is a postdoctoral researcher of the FWO (Research Foundation Flanders). The third author has been partially supported by the Spanish Government under Grant MTM2016-77445-P with “Fondos FEDER” and by Fundación Séneca of Murcia under Grant 19880/GERM/15.

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