Zassenhaus Conjecture (ZC1) on torsion units of integral group rings for some metabelian groups

Río, Ángel del; Sehgal, Sudarshan K.

doi:10.1007/s00013-005-1554-0

Arch. Math.

2006

DOI: 10.1007/s00013-005-1554-0

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Zassenhaus Conjecture (ZC1) on torsion units of integral group rings for some metabelian groups

Ángel del Río

Sudarshan K. Sehgal

Abstract: Abstract. We prove a conjecture of Zassenhaus that every normalized torsion unit of the integral group ring ZG of a finite group G is rationally conjugate to a group element for some metabelian groups including metacyclic groups G containing a normal cyclic group A such that G/A is cyclic of prime power order. The relative prime case was done in [11].

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Cited by 3 publications

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A counterexample to the first Zassenhaus conjecture

Eisele

Margolis

2018

Advances in Mathematics

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Hans J. Zassenhaus conjectured that for any unit u of finite order in the integral group ring of a finite group G there exists a unit a in the rational group algebra of G such that a −1 · u · a = ±g for some g ∈ G. We disprove this conjecture by first proving general results that help identify counterexamples and then providing an infinite number of examples where these results apply. Our smallest example is a metabelian group of order 2 7 · 3 2 · 5 · 7 2 · 19 2 whose integral group ring contains a unit of order 7 · 19 which, in the rational group algebra, is not conjugate to any element of the form ±g.

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A counterexample to the first Zassenhaus conjecture

Eisele

Margolis

2018

Advances in Mathematics

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Torsion Units in Integral Group Rings of Certain Metabelian Groups

Hertweck¹

2008

Proceedings of the Edinburgh Mathematical Society

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Zassenhaus Conjecture for cyclic-by-abelian groups

Caicedo

Margolis

Río

2013

Journal of the London Mathematical Society

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Zassenhaus Conjecture for torsion units states that every augmentation one torsion unit of the integral group ring of a finite group G is conjugate to an element of G in the units of rational group algebra QG. This conjecture has been proved for nilpotent groups, metacyclic groups and some other families of groups. We prove the conjecture for cyclic-by-abelian groups.In this paper G is a finite group and RG denotes the group ring of G with coefficients in a ring R. The units of RG of augmentation one are usually called normalized units. In the 1960s Hans Zassenhaus established a series of conjectures about the finite subgroups of normalized units of ZG. Namely he conjectured that every finite group of normalized units of ZG is conjugate to a subgroup of G in the units of QG. These conjecture is usually denoted (ZC3), while the version of (ZC3) for the particular case of subgroups of normalized units with the same cardinality as G is usually denoted (ZC2). These conjectures have important consequences. For example, a positive solution of (ZC2) implies a positive solution for the Isomorphism and Automorphism Problems (see [Seh93] for details). The most celebrated positive result for Zassenhaus Conjectures is due to Weiss [Wei91] who proved (ZC3) for nilpotent groups. However Roggenkamp and Scott founded a counterexample to the Automorphism Problem, and henceforth to (ZC2) (see [Rog91] and [Kli91]). Later Hertweck [Her01] provided a counterexample to the Isomorphism Problem.The only conjecture of Zassenhaus that is still up is the version for cyclic subgroups namely:Zassenhaus Conjecture for Torsion Units (ZC1). If G is a finite group then every normalized torsion unit of ZG is conjugate in QG to an element of G. Besides the family of nilpotent groups, (ZC1) has been proved for some concrete groups [BH08, BHK04, HK06, LP89, LT91, Her08b], for groups having a Sylow subgroup with an abelian complement [Her06], for some families of cyclic-by-abelian groups [LB83, LT90, LS98, MRSW87, PMS84, PMRS86, dRS06, RS83] and some classes of metabelian groups not necessarily cyclic-by-abelian [MRSW87, SW86]. Other results on Zassenhaus Conjectures can be found in [Seh93, Seh01] and [Seh03, Section 8]The latest and most general result for (ZC1) on the class of cyclic-by-abelian groups is due to Hertweck [Her08a] who proved (ZC1) for finite groups of the form G = AX with A a cyclic normal subgroup of G and X an abelian subgroup of

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Zassenhaus Conjecture (ZC1) on torsion units of integral group rings for some metabelian groups

Cited by 3 publications

References 14 publications

A counterexample to the first Zassenhaus conjecture

A counterexample to the first Zassenhaus conjecture

Torsion Units in Integral Group Rings of Certain Metabelian Groups

Zassenhaus Conjecture for cyclic-by-abelian groups

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