## 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

Zbl 0788.16019
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