Geodesic
On the First Zassenhaus Conjecture and Direct Products
Canadian journal of mathematics, Tome 72 (2020) no. 3, pp. 602-624

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/S0008414X18000044
Mots-clés : integral group ring, torsion unit, Zassenhaus Conjecture, direct product, Frobenius group, HeLP method 
Bächle, Andreas; Kimmerle, Wolfgang; Serrano, Mariano. On the First Zassenhaus Conjecture and Direct Products. Canadian journal of mathematics, Tome 72 (2020) no. 3, pp. 602-624. doi: 10.4153/S0008414X18000044
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