\(SK_ 1\) of finite group rings. V
[For a review of Part IV [Proc. Lond. Math. Soc., III. Ser. 46, 1--37 (1983)] see Zbl 0464.16022.] The main results of the paper are the following. The description of \(Cl_ 1(\mathbb ZG)=\text{Ker}(SK_ 1(\mathbb ZG)\to \sum_{p\mid | G|}SK_ 1({\hat {\mathbb Z}}_ pG))\) found by \textit{R. Alperin}, \textit{R. Dennis}, the author and \textit{M. Stein} [Invent. Math. 87, 253--302 (1987; Zbl 0605.18006)] for an abelian finite group \(G\) is extended here to the non-abelian \(p\)-groups (\(p\) odd). The proof gives rise to a combinatorial algorithm for the calculation of \(Cl_ 1(\mathbb ZG)\), and examples of its application for some non-abelian groups of order \(p^ 4\) are given. Since for any prime \(p\) and finite \(G\) \(Cl_ 1(\mathbb ZG)_{(p)}\) is embedded in the localization exact sequence \[ K_ 2^{top}({\hat{\mathbb Z}}_ pG)\to C_ p(\mathbb QG)\to Cl_ 1(\mathbb ZG)_{(p)}\to 0 \] whose mittle term was determined by \textit{A. Bak} and \textit{U. Rehmann} [J. Algebra 78, 475--547 (1982; Zbl 0495.20022)] the explicit description of \(Cl_ 1(\mathbb ZG)\) depends on finding generators for \(K_ 2^{top}({\hat{\mathbb Z}}_ pG)\). The author describes \(\text{Ker}(K_ 2({\hat{\mathbb Z}}_ p\alpha))\) and \(\text{Coker}(K_ 2({\hat {\mathbb Z}}_ p\alpha))\) where \(\alpha\) is a central extension of \(p\)-groups with cyclic kernel which enables to construct inductively a set of generators for \(K_ 2^{top}(\mathbb Z_ pG)^+\) when \(p\) is odd, and to compute \(Cl_ 1(\mathbb ZG)\) as above. Let \(R\) be the ring of integers in some number field. In conclusion the relationship between \(Cl_ 1(RG)\) and the complex Artin cokernel \(A_ C(G)\) is studied and the isomorphism \(Cl_ 1(RG)=A_ C(G)\) is proved for \(R\) large enough.
scientific article; zbMATH DE number 4019268