×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] G. Karpilovsky, Commutative group algebras , Marcel Dekker, New York, 1983. · Zbl 0508.16010
[2] W. May, Invariants for commutative group algebras , Illinois J. Math. 15 (1971), 525-531. · Zbl 0215.36801
[3] ——–, Isomorphism of group algebras , J. Algebra 40 (1976), 10-18. · Zbl 0329.20002 · doi:10.1016/0021-8693(76)90083-1
[4] ——–, Modular group algebras of simply presented abelian groups , Proc. Amer. Math. Soc. 104 (1988), 403-409. JSTOR: · Zbl 0691.20008 · doi:10.2307/2046986 · links.jstor.org
[5] W. Ullery, Isomorphism of group algebras , Comm. Algebra 14 (1986), 767-785. · Zbl 0587.16011 · doi:10.1080/00927878608823334
[6] ——–, A conjecture relating to the isomorphism problem for commutative group algebras , in Group and semigroup rings , North-Holland Math. Studies No. 126, North-Holland, Amsterdam, 1986, 247-252. · Zbl 0596.16012
[7] ——–, Modular group algebras of \(N\)-groups , Proc. Amer. Math. Soc. 103 (1988), 1053-1057. JSTOR: · Zbl 0656.20059 · doi:10.2307/2047084 · links.jstor.org
[8] ——–, Modular group algebras of isotype subgroups of totally projective \(p\)-groups , Comm. Algebra 17 (1989), 2325-2332. · Zbl 0683.20011 · doi:10.1080/00927878908823850
[9] ——–, An isomorphism theorem for commutative modular group algebras , Proc. Amer. Math. Soc., · Zbl 0712.20036 · doi:10.2307/2048068
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.