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Units of commutative modular group algebras. (English) Zbl 0806.16033

Let \(FG\) be the group algebra of an abelian \(p\)-group \(G\) over a perfect field \(F\) of characteristic \(p\). The direct factor problem is whether \(G\) is a direct factor of the group V(G) of the normalized units of the unit group \(U(G)\) of \(FG\). This problem has a positive solution if \(V(G)/G\) is simply presented. In the paper under review the author proves that \(V(G)/G\) has a \(\nu\)-basis. Recently the author has established that any abelian \(p\)-group \(A\) with a \(\nu\)-basis such that \(| A| \leq \aleph_ 1\) is simply presented. It is still an open problem whether a group \(A\) with a \(\nu\)-basis is simply presented if \(| A | > \aleph_ 1\).
Reviewer: T.Mollov (Plovdiv)

MSC:

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