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\(C_\lambda\)-groups and \(\lambda\)-basic subgroups in modular group rings. (English) Zbl 0989.16019
For a fixed limit ordinal \(\lambda\), denote by \(C_\lambda\) the class of all Abelian \(p\)-groups \(G\) such that \(G/G^{p^\alpha}\) is totally projective for all \(\alpha<\lambda\). C. Megibben [TĂ´hoku Math. J., II. Ser. 22, 347-356 (1970; Zbl 0222.20017)] introduced the concept of a \(\lambda\)-basic subgroup of a \(C_\lambda\)-group.
Let \(V(RG)\) be the group of normalized units in a commutative group ring \(RG\) of characteristic \(p\). The author investigates the questions, when \(V(RG)\) is a \(C_\lambda\)-group and a subgroup \(B\) of \(V(RG)\) is a \(\lambda\)-basic subgroup. In the paper answers to these questions are obtained for the case when \(G\) is an Abelian \(p\)-group, \(R\) is a perfect ring and \(\lambda\) is a countable limit ordinal. If \(V(RG)\) is a \(C_\lambda\)-group, then \(G\) is a direct factor of \(V(RG)\) with a totally projective complement.

MSC:
16U60 Units, groups of units (associative rings and algebras)
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16S34 Group rings
20K10 Torsion groups, primary groups and generalized primary groups
20K27 Subgroups of abelian groups
20K21 Mixed groups
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