Localization of q-abelian groups
type: Item
instance of: scholarly article
title: Localization of q-abelian groups
zbMATH Open document ID: 0658.20018
DOI: 10.1007/BF01324717
author: Carles Casacuberta
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: A group G is called n-abelian if \((xy)^ n=x^ ny^ n\) for all \(x,y\in G\). As first exploited by \textit{R. Baer} [in Proc. Am. Math. Soc. 4, 15-26 (1953; Zbl 0050.022)] many concepts and constructions can be generalized from abelian groups to n-abelian groups. The paper shows that, if q is a prime and G is a q-abelian group, then a universal homomorphism from G to a group in which qth roots exist and are unique can be obtained by taking the direct limit in the system \(G\to^{q}G\to^{q}G\to^{q}...\), exactly as in the abelian case. The key point in the proof is an argument which decides in general whether a given P-localization functor in a variety of groups agrees with the restriction of the P-localization functor in the whole category of all groups. As preparatory material, the paper contains several new results on the structure of n-abelian groups (together with a proof of the fact that the elements of finite order form a subgroup) and a practical procedure to obtain nontrivial examples of noncommutative n-abelian groups.
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20F45
Mathematics Subject Classification ID: 55P60
zbMATH DE Number: 4075384
zbMATH Keywords: n-abelian groups
zbMATH Keywords: qth roots
zbMATH Keywords: direct limit
zbMATH Keywords: P-localization functor
zbMATH Keywords: variety of groups
zbMATH Keywords: elements of finite order
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01324717
OpenAlex ID: W2043398386
description: scientific article; zbMATH DE number 4075384