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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The direct decompositions of a group $ G$ with $ G/G'$ finitely generated

Author: Francis Oger
Journal: Trans. Amer. Math. Soc. 347 (1995), 1997-2010
MSC: Primary 20E34; Secondary 20E07, 20F18
MathSciNet review: 1282895
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Abstract: We consider the class $ \mathcal{C}$ which consists of the groups $ M$ with $ M/M'$ finitely generated which satisfy the maximal condition on direct factors. It is well known that any $ \mathcal{C}$-group has a decomposition in finite direct product of indecomposable groups, and that two such decompositions are not necessarily equivalent up to isomorphism, even for a finitely generated nilpotent group. Here, we show that any $ \mathcal{C}$-group has only finitely many nonequivalent decompositions. In order to prove this result, we introduce, for $ \mathcal{C}$-groups, a slightly different notion of decomposition, that we call $ J$-decomposition; we show that this decomposition is necessarily unique. We also obtain, as consequences of the properties of $ J$-decompositions, several generalizations of results of R. Hirshon. For instance, we have $ \mathbb{Z} \times G \cong \mathbb{Z} \times H$ for any groups $ G$, $ H$ which satisfy $ M \times G \cong M \times H$ for a $ \mathcal{C}$-group $ M$.

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Keywords: Decompositions in direct products of indecomposable groups, cancellable in direct products, regular, maximal condition on direct factors
Article copyright: © Copyright 1995 American Mathematical Society