Isomorphism of commutative modular group algebras. (English) Zbl 0977.20003

Summary: Let \(K\) be a field of characteristic \(p>0\) and let \(G\) be a direct sum of cyclic groups, such that its torsion part is a \(p\)-group. If there exists a \(K\)-isomorphism \(KH\cong KG\) for some group \(H\), then it is shown that \(H\cong G\).
Let \(G\) be a direct sum of cyclic groups, a divisible group or a simply presented torsion Abelian group. Then \(KH\cong KG\) as \(K\)-algebras for all fields \(K\) and some group \(H\) if and only if \(H\cong G\).


20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16S34 Group rings
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