Zassenhaus Conjecture for cyclic-by-abelian groups

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 the rational group algebra QG. This conjecture has been proved for nilpotent groups, metacyclic groups and some other families of groups. It has been also proved for some special 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 1960’s 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. This conjecture is usually denoted (ZC3), while the version of (ZC3) for the particular case of subgroups of normalized units of the same order 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 discovered 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 regarding torsion units of group rings that is still open 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 normal 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]. This paper has been our main source of ideas and inspiration. Hertweck proves (ZC1) for finite groups of the form G = AX with A a cyclic normal subgroup of G and X an abelian subgroup of G. This includes the class of metacyclic groups, which was not covered in previous results. It is well known that (ZC1) holds for G if and only if the partial augmentations of torsion units are non-negative. In the hypothesis of [Her08a], CG(A) = ACX(A) = AZ(G). In the words of Hertweck this is “the main reason for assuming that A is covered by an abelian 2000 Mathematics Subject Classification 16U60, 16S34; Secondary 20C05.