On pro-\(p\)-groups with a single defining relator
type: Item
instance of: scholarly article
title: On pro-\(p\)-groups with a single defining relator
zbMATH Open document ID: 0790.20046
DOI: 10.1007/BF02801571
author: Nikolai S. Romanovskiy
published in: Israel Journal of Mathematics
publication date: 1993-06-29T00:00:00Z
review text: Let \(F\) be a free pro-\(p\)-group with a basis \(X\) and \(G = \langle X\mid r\rangle\) be a pro-\(p\)-group with a single defining relator \(r\in F^{(k)} \setminus F^{(k+1)}\), \(F^{(k)}\) be the \(n\)-th commutator subgroup of \(F\). It is proved that the factors of the derived series of \(G\) are torsion free iff \(r\) is not a \(p\)-th power in \(F^{(k)} / F^{(k+1)}\), in which case the group algebra \(\mathbb{Z}_ pG\) is a domain and \(\text{cd }G \leq 2\).
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20F14
zbMATH DE Number: 219367
zbMATH Keywords: cohomological dimension
zbMATH Keywords: free pro-\(p\)-group
zbMATH Keywords: defining relator
zbMATH Keywords: derived series
zbMATH Keywords: torsion free
zbMATH Keywords: group algebra
reviewed by: Yurii V. Mukhin
MaRDI profile type: Publication
cites work: On pro-p-groups with a single defining relator
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description: scientific article; zbMATH DE number 219367