Active sums. I.
type: Item
instance of: scholarly article
title: Active sums. I.
zbMATH Open document ID: 1069.20046
author: Alejandro Díaz-Barriga
author: F. González-Acuña
author: Francisco Marmolejo
author: Leopoldo Román
published in: Revista Matemática Complutense
publication date: 2004-10-25T00:00:00Z
full work available at URL: https://eudml.org/doc/44380
review text: Let \(U=\bigcup_iF_i\) be the disjoint union of a generating family \({\mathcal F}=\{F_i\}\) of subgroups of a group \(G\), closed under conjugation and with partial order compatible with inclusion. Define two binary operations on \(U\): partial multiplication on \(U\) induced by the multiplication in the subgroups and a global operation induced by conjugation in \(U\), \(x^y=y^{-1}xy\in F_k\) if \(x\in F_i\), \(y\in F_j\) and \(F^y_i=F_k\). Then \(U\) is a partial algebra called an active add determined by \(\mathcal F\). This active add determines a group \(S\) which is the image of \(U\) under left adjoint to the inclusion functor from the category of groups to the category of active adds. \(S\) is called the active sum of \(U\). The authors show that \(\varphi^{-1}(Z(G))=Z(S)\), where \(\varphi\) is the natural epimorphism from \(S\) onto \(G\) and \(Z\) denotes the center. They basically investigate when is the active sum \(S\) of the family \(\mathcal F\) isomorphic to the group \(G\) and show that the following groups are active sums of cyclic subgroups: free groups, groups of order cube of an odd prime, simple groups with trivial Schur multiplier, and special linear groups. They also show that every finite group \(G\) with \(G/G'\) non-cyclic is the active sum of proper normal subgroups.
reviewed by: C. G. Chehata
Mathematics Subject Classification ID: 20J15
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20D99
Mathematics Subject Classification ID: 08A55
zbMATH DE Number: 2109938
zbMATH Keywords: active sums of cyclic groups
zbMATH Keywords: partial algebras
zbMATH Keywords: regularity
zbMATH Keywords: independence
zbMATH Keywords: atomic groups
zbMATH Keywords: molecular groups
zbMATH Keywords: categories of groups
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2109938