On the unsolvability of the conjugacy problem for subgroups of the group \(R_5\) of pure braids
type: Item
instance of: scholarly article
title: On the unsolvability of the conjugacy problem for subgroups of the group \(R_5\) of pure braids
zbMATH Open document ID: 0952.20027
DOI: 10.1007/BF02675004
published in: Mathematical Notes
publication date: 2001-01-14T00:00:00Z
review text: The braid group \(B_{n+1}\) is defined by the generators \(\sigma_1,\dots,\sigma_n\) and the defining relations \[ \sigma_i\sigma_j\sigma_i=\sigma_j\sigma_i\sigma_j\;(|i-j|=1),\quad\sigma_i\sigma_j=\sigma_j\sigma_i\;(|i-j|>1). \] There exists a homomorphism of the group \(B_{n+1}\) to the permutation group \(S_{n+1}\) that maps the generators \(\sigma_i\) to the transpositions \((i,i+1)\). A braid that realizes the identity permutation is said to be pure. The subgroup of pure braids of the group \(B_{n+1}\) is denoted by \(R_{n+1}\). \textit{I. V. Dobrynina} [Algorithmic problems of the theory of groups and semigroups, Tul'sk. Gos. Ped. Inst., Tula 1994, 62-70 (1994)] proved the unsolvability of the conjugacy problem for subgroups of the groups \(R_{n+1}\) for \(n>4\). In this paper the authors prove the same result for \(n=4\).
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 20F10
zbMATH DE Number: 1407529
zbMATH Keywords: occurrence problem
zbMATH Keywords: braid groups
zbMATH Keywords: generators
zbMATH Keywords: relations
zbMATH Keywords: pure braids
zbMATH Keywords: conjugacy problem
author: V. N. Bezverkhniĭ
author: I. V. Dobrynina
reviewed by: Andrei M. Akimenkov
MaRDI profile type: Publication
cites work: Theory of braids
cites work: Subgroups of the braid group \(B_ 4\)
full work available at URL: https://doi.org/10.1007/bf02675004
OpenAlex ID: W2009370715
description: scientific article; zbMATH DE number 1407529