The derived group of a permutation representation
type: Item
instance of: scholarly article
title: The derived group of a permutation representation
zbMATH Open document ID: 0547.57001
DOI: 10.1016/0001-8708(84)90020-3
published in: Advances in Mathematics
publication date: 1984-01-01T00:00:00Z
review text: The author describes an algebraic construction which unifies in a single theory the basic algebra associated with knot groups, Alexander matrices, and covering spaces. The construction is modelled on the algebra of liftings of closed paths in a base space B to paths in a covering space E of B. The object constructed is a group A. The author proves a structure theorem for this group which, in the model case, exhibits the group as the free product of \(\pi_ 1(E)\) with a free group. The author uses the construction to give an elegant formulation of the Reidemeister-Schreier algorithm for finding a presentation of a subgroup of a presented group.
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 3872279
zbMATH Keywords: knot groups
zbMATH Keywords: Alexander matrices
zbMATH Keywords: covering spaces
zbMATH Keywords: algebra of liftings of closed paths
zbMATH Keywords: free product
zbMATH Keywords: free group
zbMATH Keywords: Reidemeister-Schreier algorithm
zbMATH Keywords: presentation of a subgroup
author: Richard H. Crowell
MaRDI profile type: Publication
full work available at URL: 0001-8708(84)90020-3
OpenAlex ID: W2008349536
cites work: Q5545247
cites work: Corresponding Group and Module Sequences
cites work: The derived module of a homomorphism
cites work: Q5528469
cites work: Free differential calculus. II: The isomorphism problem of groups
cites work: A quick trip through knot theory
cites work: Q5632789
cites work: Q5732691
cites work: Alexander ideals of links
cites work: Q4145882
cites work: Q5343374
description: scientific article; zbMATH DE number 3872279