Higher \(K'\)-groups of integral group rings (Q1182511)

Suppose a finite group \(G\) is the semi-direct product \(\pi\rtimes\Gamma\), where \(\pi\) is nilpotent and \(\Gamma\) is arbitrary. The author shows that \(G_ q(\mathbb{Z}[\pi\rtimes\Gamma])\) decomposes as the direct sum of \(G_ q\;(=K_ q')\) of certain twisted group rings \(\mathbb{Z}\langle\Gamma_ \rho\rangle\#\Gamma\). The direct sum is indexed by the orbits \(\Gamma_ \rho\) of the action of \(\Gamma\) on rational representations of \(\pi\), and \(\mathbb{Z}\) can in fact be replaced by an arbitrary coefficient ring \(R\) with identity. This theorem, by suitable specialization, implies earlier results of Webb and Hambleton, Taylor and Williams; it also confirms a formula conjectured by the latter three authors.