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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