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An isomorphism theorem for commutative modular group algebras. (English) Zbl 0712.20036

A new class of abelian \(p\)-groups, called \(A_n(\mu)\)-groups, is introduced in the paper where \(n\) is an arbitrary positive integer and \(\mu\) is a limit ordinal. It is shown that this class consists entirely of totally projective \(p\)-groups if and only if \(\mu\) is cofinal with \(\omega_0\). These groups are uniquely determined up to isomorphisms by numerical invariants (Theorem 2). As an application it is proved the following isomorphism result (Theorem 4): If \(F\) is a field of characteristic \(p\), \(H\) is an \(A_n(\mu)\)-group and \(K\) is a group such that the group algebras \(FH\) and \(FK\) are \(F\)-isomorphic, then \(H\) and \(K\) are isomorphic.
Reviewer: T.Mollov

MSC:

20K10 Torsion groups, primary groups and generalized primary groups
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16S34 Group rings
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References:

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