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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
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