Products of powers and commutators in free pro-p-groups
type: Item
instance of: scholarly article
title: Products of powers and commutators in free pro-p-groups
zbMATH Open document ID: 0647.20021
DOI: 10.1007/BF01159665
author: Oleg Melnikov
published in: Mathematical Notes
publication date: 1986-01-01T00:00:00Z
review text: It is known that an (abstract) group with one defining relation \(r=1\) is not a proper power. For pro-p-groups the analogous statement is false: \textit{D. Gildenhuys} [Invent. Math. 5, 357-366 (1968; Zbl 0159.306)] has constructed examples of pro-p-groups, with a single defining relator that is not a p-th power, that nevertheless contain elements of finite order. The question naturally arises of how special such examples are. It is answered by the following Proposition: Every noncyclic pro-p-group with generators \(x_ i\), \(i\in I\), and single defining relation can be presented by a single defining relation \(r=1\), where r is not a proper power (in the free pro-p-group on \(\{x_ i|\) \(i\in I\})\).
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F12
zbMATH DE Number: 4055878
zbMATH Keywords: pro-p-groups
zbMATH Keywords: relator
zbMATH Keywords: generators
zbMATH Keywords: free pro-p-group
MaRDI profile type: Publication
cites work: On pro-p-groups with a single defining relator
cites work: Residual nilpotence and relations in free groups
cites work: A subgroup theorem for free products of pro-finite groups
cites work: On free nilpotent quotient groups
cites work: Produkte von Potenzen und Kommutatoren in freien Gruppen
description: scientific article; zbMATH DE number 4055878