The coherence number of a finite group
type: Item
instance of: scholarly article
title: The coherence number of a finite group
zbMATH Open document ID: 0699.55007
DOI: 10.1016/0021-8693(89)90320-7
author: William J. Ralph
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: The algebraic problem we will identify arose within algebraic topology in the course of the development of a transfer for homotopy theory. This transfer can be applied to the action of finite groups on topological spaces. More specifically, to each finite group G the theory associates, in a natural way, a distinguished element \(\theta_ G\) in a subgroup \(T_ G\) of a direct product of free groups. In order to apply the transfer, one is first required to compute the order of the image of \(\theta_ G\) in the abelianization, \(T_ G^{ab}\), of the subgroup \(T_ G\). It is this order, possibly infinite, which we have labelled as being the ``coherence number'' of the group G. As we will show, \(T_ G^{ab}\) is isomorphic to \({\mathbb{Z}}_{c(G)}\oplus {\mathbb{Z}}^{2| G| -1}\), so the group \(T_ G^{ab}\) is completely determined by the coherence number and order of G. As we will show, there are some delicate connections between groups and their subgroups and their respective coherence numbers. Fortunately it is true that the coherence number of a subgroup divides the coherence number of the group, but the situation for quotient groups is more subtle. The best that we have done so far is to show that the coherence number of the quotient of G by a normal subgroup H divides the product of the order of H and the coherence number of G. It is unknown in general whether the coherence number of a group is always finite. In general, it seems that it will be very large.
Mathematics Subject Classification ID: 55P91
zbMATH DE Number: 4147096
zbMATH Keywords: transfer
zbMATH Keywords: coherence number
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90320-7
OpenAlex ID: W2046699028
cites work: Q3866809
cites work: Category and Group Rings in Homotopy Theory
cites work: An extension of singular homology to Banach algebras
cites work: Q4858090
Recommended article: The Coherence Number of 2-Groups
Recommended article: Finite Abelian group cohesion
Recommended article: Q4303983
Recommended article: Q4696633
Recommended article: Finite co-Dedekindian groups
Recommended article: Cohomology of finite soluble groups
Recommended article: Q5673073
Recommended article: The Coexponent of a Finitep-Group
Recommended article: Q4439201
Recommended article: The order of a finite Coxeter group
description: scientific article; zbMATH DE number 4147096