## Basic subgroups of the group of normalized units of modular group rings.(English)Zbl 0859.16025

Let $$G$$ be an abelian $$p$$-group and let $$R$$ be a commutative ring with identity of characteristic $$p$$. This ring $$R$$ is called $$p$$-perfect if $$R^p=R$$, where $$R^p=\{x^p\mid x\in R\}$$. Let $$V(RG)$$ be the group of normalized units of the group ring $$RG$$. Obviously $$V(RG)$$ coincides with its $$p$$-component $$S(RG)$$. If $$H$$ is a subgroup of $$G$$ then let $$I(RG,H)$$ be the ideal of the group ring $$RG$$ generated by the elements $$h-1$$, $$h\in H$$ and let $$S(RG;H)=1+I(RG,H)$$.
The main result of the paper is the following: if the ring $$R$$ is $$p$$-perfect and if $$B$$ is a basic subgroup of $$G$$ then $$S(RG;B)$$ is a basic subgroup of $$S(RG)$$. An example is given when not every basic subgroup of $$S(RG)$$ can be represented in the way mentioned above.
Reviewer: T.Mollov (Plovdiv)

### MSC:

 16U60 Units, groups of units (associative rings and algebras) 16S34 Group rings 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 20E07 Subgroup theorems; subgroup growth