On genus and embeddings of torsion-free nilpotent groups of class two |
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Authors: | Carles Casacuberta Charles Cassidy Dirk Scevenels |
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Institution: | (1) Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain;(2) Département de Mathématiques et de Statistique, Université Laval, G1K 7P4, Québec, Canada;(3) Department Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3001 Heverlee, Belgium;(4) Present address: CRM, Apartat 50, E-08193 Bellaterra, Spain |
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Abstract: | Summary We study embeddings between torsion-free nilpotent groups having isomorphic localizations. Firstly, we show that for finitely
generated torsion-free nilpotent groups of nilpotency class 2, the property of having isomorphicP-localizations (whereP denotes any set of primes) is equivalent to the existence of mutual embeddings of finite index not divisible by any prime
inP. We then focus on a certain family Γ of nilpotent groups whose Mislin genera can be identified with quotient sets of ideal
class groups in quadratic fields. We show that the multiplication of equivalence classes of groups in Γ induced by the ideal
class group structure can be described by means of certain pull-back diagrams reflecting the existence of enough embeddings
between members of each Mislin genus. In this sense, the family Γ resembles the family N0 of infinite, finitely generated nilpotent groups with finite commutator subgroup. We also show that, in further analogy with
N0, two groups in Γ with isomorphic localizations at every prime have isomorphic localizations at every finite set of primes.
We supply counterexamples showing that this is not true in general, neither for finitely generated torsion-free nilpotent
groups of class 2 nor for torsion-free abelian groups of finite rank.
Supported by DGICYT grant PB94-0725
This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag. |
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