Group algebras with centrally metabelian unit groups
The paper determines the finite groups \(G\) having the property that the unit group \(U\) of their group algebra \(KG\) over a field \(K\) of positive characteristic \(\neq 2\) is centrally metabelian, meaning that the second derived term \(\delta^2U\) is central in \(U\). This holds, for example, whenever \(KG\) is Lie centrally metabelian (\(G\) may even be infinite), and \textit{M. Sahai} and \textit{J. B. Srivastava} [J. Algebra 187, No. 2, 7-15 (1997)]; \textit{R. K. Sharma} and \textit{J. B. Srivastava} [J. Algebra 151, 476-486 (1992; Zbl 0788.16019)] have provided necessary and sufficient conditions for \(KG\) being so. In the paper under review the full list of groups \(G\) is shown to consist of all abelian \(G\) and, moreover, in characteristic 3, groups \(G\) with commutator subgroup of order 3. The proofs are rather computational.
scientific article; zbMATH DE number 999994