Zassenhaus Conjecture for cyclic-by-abelian groups

Abstract

Zassenhaus Conjecture for torsion units states that every aug- mentation 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 ringR. 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 sub- groups namely: