The torsion type of nilpotent groups
type: Item
instance of: scholarly article
title: The torsion type of nilpotent groups
zbMATH Open document ID: 0819.20037
author: Hans Peter Heislbetz
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1994-09-22T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108329
review text: In the theory of torsion-free abelian groups of finite rank the outer type of such groups plays an important role. This type can be defined from certain torsion quotients. In this paper this definition is extended to torsion-free nilpotent groups \(G\) of finite rank. Many properties of this invariant type are exhibited and the differences from the abelian case are pointed out. There is a connection between the torsion-types of \(G/\zeta_{k-1}(G)\) and \(\gamma_ k(G)\), where \(\zeta_ i(G)\) denotes the upper central series and \(\gamma_ i(G)\) the lower central series of \(G\). Moreover, a formula for the torsion type of a product \(UN\) of a subgroup \(U\) and a normal subgroup \(N\) of a nilpotent group of finite rank is given. The special commutator identities that are needed in the proofs for the nilpotent case are perhaps of independent interest.
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20K15
zbMATH DE Number: 627426
zbMATH Keywords: torsion-free abelian groups of finite rank
zbMATH Keywords: outer types
zbMATH Keywords: torsion-free nilpotent groups
zbMATH Keywords: upper central series
zbMATH Keywords: lower central series
zbMATH Keywords: torsion types
zbMATH Keywords: commutator identities
reviewed by: Bernhard Amberg
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 627426