Idempotent transformations of finite groups.
type: Item
instance of: scholarly article
title: Idempotent transformations of finite groups.
zbMATH Open document ID: 1293.20051
author: M. Blomgren
author: Wojciech Chachólski
author: Emmanuel D. Farjoun
author: Yoav Segev
published in: Advances in Mathematics
publication date: 2013-01-16T00:00:00Z
review text: Let \(\mathcal G\) be the category of groups and \(\Phi\colon\mathcal G\rightsquigarrow\mathcal G\) be an augmented functor that is an endofunctor with a natural transformation \(\varepsilon\colon\Phi\rightsquigarrow\mathrm{id}\), called augmentation. An augmented functor \((\Phi,\varepsilon)\) is called idempotent if the two homomorphisms \(\Phi(\varepsilon(G))\) and \((\Phi(G))\colon\Phi^2(G)\rightsquigarrow\Phi(G)\) are isomorphisms for any group \(G\). The aim of this paper is to understand how idempotent functors deform (finite) groups. The authors show that finiteness, nilpotency, solvability,\dots are preserved by idempotent functors. Let \(\Phi\) be an idempotent functor and let \(G\) be a finite group, then the authors prove that the set \(\mathrm{idem}(G)\) of isomorphism classes of \(\Phi(G)\) is finite. They enumerate the set \(\mathrm{idem}(G)\) and study the relation between idempotent functors and cellular cover. At the end, the authors give explicit examples to illustrate some results of the paper.
Mathematics Subject Classification ID: 20J15
Mathematics Subject Classification ID: 18A22
Mathematics Subject Classification ID: 55P60
Mathematics Subject Classification ID: 55M35
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 6126014
zbMATH Keywords: categories of groups
zbMATH Keywords: idempotent functors
zbMATH Keywords: finite simple groups
zbMATH Keywords: cellular covers
reviewed by: Saïd Zarati
MaRDI profile type: Publication
full work available at URL: j.aim.2012.09.030
OpenAlex ID: W2094900065
DOI: 10.1016/J.AIM.2012.09.030
description: scientific article; zbMATH DE number 6126014