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Isomorphism of commutative group algebras of \(p\)-mixed splitting groups over rings of characteristic zero. (English) Zbl 1111.20007

Summary: Suppose \(G\) is a \(p\)-mixed splitting Abelian group and \(R\) is a commutative unitary ring of characteristic zero such that the prime number \(p\) satisfies \(p\notin\text{inv}(R)\cup\text{zd}(R)\). Then \(R(H)\) and \(R(G)\) are canonically isomorphic \(R\)-group algebras for any group \(H\) precisely when \(H\) and \(G\) are isomorphic groups.

MSC:

20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16S34 Group rings
16U60 Units, groups of units (associative rings and algebras)
20K21 Mixed groups
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