K₁ of a p-adic group ring II. The determinantal kernel SK₁

We describe the group SK_1(R[G]) for group rings R[G] where G is an arbitrary finite group and where the coefficient ring R is a p-adically complete Noetherian integral domain of characteristic zero which admits a lift of Frobenius and which also satisfies a number of further mild conditions. Our results extend previous work of R. Oliver who obtained such results for the valuation rings of finite extensions of the p-adic field.