Zassenhaus conjecture for cyclic-by-Abelian groups. (Q2843981)

Three celebrated 50 years old conjectures on units in group rings over a finite group \(G\) are known as the Zassenhaus conjectures. The first conjecture (ZC1) is still open and states that every augmentation 1 torsion unit of \(\mathbb ZG\) is conjugate within \(\mathbb QG\) to an element of \(G\). For nilpotent groups, a stronger statement (ZC3) was proved in 1991 by A. Weiss. Hertweck (2008) has proved (ZC1) for a class of cyclic-by-Abelian groups, namely the finite groups \(G=AX\) with cyclic normal \(A\) and \(X\) Abelian. In the present paper, Hertweck's ideas are extended in a substantial way to obtain (ZC1) for all cyclic-by-Abelian groups \(G\).