The automorphism groups of relatively free algebras
type: Item
instance of: scholarly article
title: The automorphism groups of relatively free algebras
zbMATH Open document ID: 0919.20019
author: Roger M. Bryant
author: Vitaliĭ Roman'kov
published in: Journal of Algebra
publication date: 1999-09-08T00:00:00Z
review text: The authors study the automorphisms groups of relatively free algebras having in mind the automorphism groups of relatively free groups. They saw that the results can be formulated by considering relatively free algebras of infinite rank. So let \(\Omega\) be an operator domain. An \(\Omega\)-algebra is a set \(G\) in which for every \(\omega\in\Omega\) there is an \(n\)-ary operation of \(G\) for some integer \(n\). If \(X\) is a subset of \(G\), then \(G\) is called relatively free on \(X\), and \(X\) a free generating set for \(G\), if \(X\) generates \(G\) and every function \(X\to G\) can be extended to an \(\Omega\)-algebra homomorphism \(G\to G\). If \(U\) is a subset of \(\Gamma=\Aut(G)\), then \(\Gamma_{(U)}\) denotes the subgroup of \(\Gamma\) consisting of the pointwise stabilizers of \(U\) and \(\Gamma\) becomes a topological group where the open sets are the unions of cosets of subgroups of this form with \(U\) finite. The authors prove the existence of large dense free subgroups of \(\Gamma\) for \(G\) of infinite rank \(| X|=\kappa\). If \(G\) is an \(\Omega\)-algebra, then \(G\) is said to have the small index property if every subgroup of \(\Aut(G)\) with index less than \(2^{\aleph_0}\) is open. We can quote, with no further terminology, two of the main theorems of this paper: Let \(\Omega\) be an operator domain and let \(G\) be a relatively free \(\Omega\)-algebra of infinite rank \(\kappa\). Suppose that \(| G|\leq 2^\kappa\) and let \(\eta\) be any cardinal such that \(| G|\leq\eta\leq 2^\kappa\). Then \(\Aut(G)\) has a dense subset of cardinality \(\eta\) which freely generates a free subgroup of \(\Aut(G)\). Let \(\Omega\) be an operator domain and let \(G\) be a relatively free \(\Omega\)-algebra of infinite rank. If \(G\) has the small index property then \(\Aut(G)\) has no proper normal subgroup of index less than \(2^{\aleph_0}\). Hence \(\Aut(G)\) is perfect.
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20B27
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 08B20
Mathematics Subject Classification ID: 08A35
zbMATH DE Number: 1247624
zbMATH Keywords: automorphism groups
zbMATH Keywords: relatively free algebras
zbMATH Keywords: relatively free groups
zbMATH Keywords: free subgroups
zbMATH Keywords: small index property
MaRDI profile type: Publication
full work available at URL: jabr.1998.7559
OpenAlex ID: W2054203623
cites work: The Small Index Property for Free Groups and Relatively Free Groups
cites work: On automorphisms of relatively free groups
cites work: Automorphisms of free metabelian groups of infinite rank
cites work: Automorphisms of free nilpotent-by-abelian groups
cites work: Automorphisms of relatively free nilpotent groups of infinite rank
cites work: Automorphism groups of relatively free groups
cites work: Q5515373
cites work: Q3246392
cites work: Most Finitely Generated Permutation Groups are Free
cites work: The Small Index Property for ω‐Stable ω‐Categorical Structures and for the Random Graph
cites work: The Automorphism Group of a Countable Recursively Saturated Structure
cites work: Q4309351
cites work: Autour De La Propriété Du Petit Indice
cites work: GROUPS OF AUTOMORPHISMS OF N0-CATEGORICAL STRUCTURES
cites work: Aut (M ) Has a Large Dense Free Subgroup for Saturated M
DOI: 10.1006/JABR.1998.7559
description: scientific article; zbMATH DE number 1247624