On \(t\)-uniform groups
type: Item
instance of: scholarly article
title: On \(t\)-uniform groups
zbMATH Open document ID: 0817.20041
DOI: 10.1007/BF01201448
author: Hans Peter Heislbetz
published in: Archiv der Mathematik
publication date: 1994-09-22T00:00:00Z
review text: A group \(G\) is said to be hyperrational if it has an ascending series \(1 = G_ 0 \leq G_ 1 \leq \cdots \leq G_ \alpha\) such that, for every \(\beta < \alpha\), \(G_ \beta\) is an isolated normal subgroup of \(G_{\beta + 1}\) and the factor group \(G_{\beta + 1}/G_ \beta\) is isomorphic to a subgroup of the additive group of rational numbers. If the above series can be chosen to be finite, the group \(G\) is called polyrational. Moreover, the hyperrational group \(G\) is called \(t\)-uniform if all the factor groups \(G_{\beta + 1}/G_ \beta\) have the same type \(t\). Among other results, it is proved that if \(G\) is a non-abelian \(t\)- uniform Engel group, then the type \(t\) is idempotent. It follows that if \(G\) is a \(t\)-uniform polyrational non-abelian \(R\)-group, then \(t\) is idempotent (here \(G\) is said to be an \(R\)-group if for every positive integer \(n\) and for all elements \(x, y \in G\) such that \(x^ n = y^ n\) we have \(x = y\)). Moreover, some results on homogeneous \(t\)-uniform groups and on normal homogeneous subgroups are obtained.
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20F45
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 493391
zbMATH Keywords: ascending series
zbMATH Keywords: isolated normal subgroups
zbMATH Keywords: additive group of rational numbers
zbMATH Keywords: hyperrational groups
zbMATH Keywords: non-abelian \(t\)-uniform Engel groups
zbMATH Keywords: \(R\)- groups
zbMATH Keywords: homogeneous \(t\)-uniform groups
zbMATH Keywords: normal homogeneous subgroups
MaRDI profile type: Publication
cites work: Some aspects of groups with unique roots
cites work: Q5609457
cites work: Q5612538
cites work: Divisibility and type-conditions in generalized nilpotent groups
cites work: Invariante Typen in torsionsfreien, auflösbaren Gruppen endlichen Ranges. (Invariant types in torsion free soluble groups of finite rank)
cites work: On Infinite Soluble Groups (IV)
cites work: Q3270631
cites work: Q3832772
cites work: Q4009774
cites work: Rank restricting properties of finitely generated soluble groups
cites work: Q5654235
cites work: Q3941589
full work available at URL: https://doi.org/10.1007/bf01201448
OpenAlex ID: W2314495667
description: scientific article; zbMATH DE number 493391