On the universality of systems of words in permutation groups
type: Item
instance of: scholarly article
title: On the universality of systems of words in permutation groups
zbMATH Open document ID: 0541.20003
author: Manfred Droste
author: S. Shelah
published in: Pacific Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let \(G\) be a group, \(x_ 1,...,x_ n\) variables, \(y_ 1,...,y_ m\) parameters, and \(w_ i=w_ i(x_ 1,...,x_ n;y_ 1,...,y_ m)\) (\(i\in I)\) group words in these variables and parameters. Then \(W=\{w_ i;\quad i\in I\}\) is called \(G\)-universal if for all \(y_ 1,...,y_ m\in G\) there exist \(x_ 1,...,x_ n\in G\) such that for all \(i\in I,\quad w_ i(x_ 1,...,x_ n;y_ 1,...,y_ m)=e.\) Two groups \(G\) and \(H\) are called equationally equivalent if for any finite set W of words \(w_ i\) as above, \(W\) is \(G\)-universal iff \(W\) is \(H\)-universal, i.e. \(G\) and \(H\) cannot be distinguished by the solvability of a system of finitely many equations in variables and parameters. In the classes of infinite symmetric groups, their normal subgroups, and their factor groups, the authors determine those groups which are equationally equivalent.
Mathematics Subject Classification ID: 20B30
Mathematics Subject Classification ID: 20B35
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 3859374
zbMATH Keywords: group words
zbMATH Keywords: equationally equivalent
zbMATH Keywords: infinite symmetric groups
zbMATH Keywords: normal subgroups
MaRDI profile type: Publication
full work available at URL: pjm.1987.127.321
OpenAlex ID: W1531072470
DOI: 10.2140/PJM.1987.127.321
description: scientific article; zbMATH DE number 3859374