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Group algebras with centrally metabelian unit groups. (English) Zbl 0869.16024

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 M. Sahai and J. B. Srivastava [J. Algebra 187, No. 2, 7-15 (1997)]; R. K. Sharma and 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.

MSC:

16U60 Units, groups of units (associative rings and algebras)
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16S34 Group rings
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20F45 Engel conditions
20F14 Derived series, central series, and generalizations for groups

Citations:

Zbl 0788.16019
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