Co-actions of groups
type: Item
instance of: scholarly article
title: Co-actions of groups
zbMATH Open document ID: 0984.20013
DOI: 10.1017/S0308210500000858
publication date: 2002-02-19T00:00:00Z
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 55P05
zbMATH DE Number: 1610110
zbMATH Keywords: homomorphisms
zbMATH Keywords: projections
zbMATH Keywords: free products
zbMATH Keywords: co-actions
zbMATH Keywords: generators
author: Martin Arkowitz
author: M. A. Gutierrez
MaRDI profile type: Publication
full work available at URL: s0308210500000858
OpenAlex ID: W2013087618
published in: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics
review text: Let \(f\colon G\to H\) be a fixed homomorphism and \(p'\colon G*H\to G\) and \(p''\colon G*H\to H\) the two projections of the free product. Then a co-action relative to \(f\) is a homomorphism \(s\colon G\to G*H\) such that \(p's=id\) and \(p''s=f\) where \(id\) is the identity homomorphism. The authors study this notion.NEWLINENEWLINENEWLINEThey show a co-action \(s\colon G\to G*H\) relative to \(f\) gives a free product decomposition of \(G\) into a free subgroup and a free product of subgroups of \(\ker f\). The authors obtain a complete description of \(s\) on all finite subgroups of \(G\). They find a canonical set of generators of \(G\) in the case when the co-action \(s\) is associative. Also, the authors investigate the action of \(\Hom(H,B)\) on \(\Hom(G,B)\) induced by \(s\colon G\to G*H\).NEWLINENEWLINENEWLINELet \(m\colon H\to H*H\) be a homomorphism such that \(p'm=p''m=id\), where \(p',p''\colon H*H\to H\) are the two projections. Suppose \(A\) is subgroup of \(H\). It is proved if \(m(A)\subseteq A*H\) then \(A\) is a free factor of \(H\). In addition, the authors give several diverse examples of co-actions.
reviewed by: Alexander Ivanovich Budkin
description: scientific article; zbMATH DE number 1610110