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On isomorphism of group algebras of torsion Abelian groups. (English) Zbl 0773.16008
Suppose $$R$$ is a commutative ring with 1, $$G$$ is a torsion Abelian group and $$RG$$ is the group algebra of $$G$$ over $$R$$. Let $$\text{inv}(R)$$ be the set of rational primes $$p$$ which invert in $$R$$ and set $$G_ R=\coprod\{G_ p:p\in\text{inv}(R)\}$$. The group $$G$$ is called $$R$$- favorable if $$G_ R$$ is the trivial subgroup of $$G$$. It is proved that the following problems are equivalent: (1) For every field $$F$$ of nonzero characteristic $$p$$ and for every $$p$$-group $$G$$, $$FG\cong FH$$ for some group $$H$$ implies that $$G\cong H$$. (2) For every ring $$R$$ of characteristic 0 and for all $$R$$-favorable torsion groups $$G$$ and $$H$$, $$RG\cong RH$$ implies that $$G\cong H$$. If the additive group of $$R$$ is assumed to be torsion free and if $$G$$ is $$R$$-favorable it is shown that the isomorphism class of $$RG$$ determines the isomorphism class of $$G$$.
Reviewer: T.Mollov (Plovdiv)

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