Descriptions of groups of prime-power order
type: Item
instance of: scholarly article
title: Descriptions of groups of prime-power order
zbMATH Open document ID: 0921.20019
author: M. F. Newman
author: Werner Nickel
author: Alice C. Niemeyer
published in: Journal of Symbolic Computation
publication date: 1999-08-09T00:00:00Z
review text: The authors describe an algorithm to construct a power-commutator presentation of the \(p\)-covering group \(G^*:=F/[F,R]R^p\) of a \(p\)-group \(G\) given by means of a power-commutator presentation where \(G\) can be realized as a factor \(F/R\) of the free group \(F\) with \(d\) generators and \(d\) minimal with this property. The covering algorithm is split into two major parts: (a) a tails routine extending the consistent power-commutator presentation of \(G\) to a non-consistent power-commutator presentation of the covering group by adding further defining relations and generators and appending tails of words in the new generators to the group relations, (b) a consistency routine which removes redundant generators to make the power-commutator presentation of the covering group consistent again. The article contains an old version of a tails routine not published elsewhere which -- for example -- constructs the covering group of \(B(4,4)\), the free group of exponent 4 on four generators, with 61,780 new generators. A new tails routine is introduced and proven correct which only adds 1,058 new generators. A minimum of 856 new generators is required to define the covering group. -- The new tails routine is also included in pseudo-code.
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20B40
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 1204293
zbMATH Keywords: \(p\)-covering groups
zbMATH Keywords: power-commutator presentations
zbMATH Keywords: tails routines
zbMATH Keywords: weighted power-commutator presentations
zbMATH Keywords: consistent power-commutator presentations
zbMATH Keywords: lower exponent-\(p\) central series
zbMATH Keywords: algorithms
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: generators
zbMATH Keywords: relations
zbMATH Keywords: Burnside groups of exponent \(4\)
MaRDI profile type: Publication
full work available at URL: jsco.1997.0193
OpenAlex ID: W2059808601
DOI: 10.1006/JSCO.1997.0193
description: scientific article; zbMATH DE number 1204293