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

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DOI

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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     title = {On the {First} {Zassenhaus} {Conjecture} and {Direct} {Products}},
     journal = {Canadian journal of mathematics},
     pages = {602--624},
     year = {2020},
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