The direct decompositions of a group with 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 which consists of the groups with finitely generated which satisfy the maximal condition on direct factors. It is well known that any -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 -group has only finitely many nonequivalent decompositions. In order to prove this result, we introduce, for -groups, a slightly different notion of decomposition, that we call -decomposition; we show that this decomposition is necessarily unique. We also obtain, as consequences of the properties of -decompositions, several generalizations of results of R. Hirshon. For instance, we have for any groups , which satisfy for a -group .

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DOI:
https://doi.org/10.1090/S0002-9947-1995-1282895-8

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