zassenhaus conjecture
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2020 ◽  
Vol 559 ◽  
pp. 529-546
Author(s):  
Dietrich Burde ◽  
Wolfgang Alexander Moens

2019 ◽  
Vol 22 (5) ◽  
pp. 953-974
Author(s):  
Ángel del Río ◽  
Mariano Serrano

Abstract H. J. Zassenhaus conjectured that any unit of finite order and augmentation 1 in the integral group ring {\mathbb{Z}G} of a finite group G is conjugate in the rational group algebra {\mathbb{Q}G} to an element of G. We prove the Zassenhaus conjecture for the groups {\mathrm{SL}(2,p)} and {\mathrm{SL}(2,p^{2})} with p a prime number. This is the first infinite family of non-solvable groups for which the Zassenhaus conjecture has been proved. We also prove that if {G=\mathrm{SL}(2,p^{f})} , with f arbitrary and u is a torsion unit of {\mathbb{Z}G} with augmentation 1 and order coprime with p, then u is conjugate in {\mathbb{Q}G} to an element of G. By known results, this reduces the proof of the Zassenhaus conjecture for these groups to proving that every unit of {\mathbb{Z}G} of order a multiple of p and augmentation 1 has order actually equal to p.


2019 ◽  
Vol 531 ◽  
pp. 320-335
Author(s):  
Leo Margolis ◽  
Ángel del Río ◽  
Mariano Serrano

2018 ◽  
Vol 339 ◽  
pp. 599-641 ◽  
Author(s):  
Florian Eisele ◽  
Leo Margolis

2018 ◽  
Vol 72 (3) ◽  
pp. 602-624
Author(s):  
Andreas Bächle ◽  
Wolfgang Kimmerle ◽  
Mariano Serrano

AbstractIn 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.


2018 ◽  
Vol 507 ◽  
pp. 292-319 ◽  
Author(s):  
Leo Margolis ◽  
Ángel del Río

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