Bounds in the theory of finite covers
type: Item
instance of: scholarly article
title: Bounds in the theory of finite covers
zbMATH Open document ID: 1003.20002
author: David M. Evans
author: Osama A. Rashwan
published in: Journal of Algebra
publication date: 2002-07-04T00:00:00Z
review text: Let \(\Gamma_0\) be the unrestricted wreath product \(F\text{ Wr}_W\text{Sym}(\Omega)\), where \(F\) is a finite nilpotent group, \(\Omega\) is an infinite set, and \(W\) is the collection of \(n\)-tuples of distinct elements from \(\Omega\) (with \(\text{Sym}(\Omega)\) acting naturally). The authors give an upper bound (in terms of \(n\) and the composition length of \(F\)) on the number of conjugacy classes of closed subgroups of \(\Gamma_0\) which project onto \(\text{Sym}(\Omega)\). The methods include an extension to infinite symmetric groups of results on representations of finite symmetric groups, some group cohomology, and a result of Krajíček on the weight of generating sets for submodules of tabloid modules for the infinite degree symmetric groups. Part of the original motivation for this was model-theoretic. Any disintegrated totally categorical structure is bi-interpretable with a finite cover of a Grassmannian of a pure set. It is known that if the arity and the fibre group of such a structure is fixed, then the number of possible covers is finite. The present paper gives bounds in certain cases.
Mathematics Subject Classification ID: 20B27
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1763559
zbMATH Keywords: finite covers
zbMATH Keywords: closed permutation groups
zbMATH Keywords: infinite symmetric groups
zbMATH Keywords: wreath products
zbMATH Keywords: numbers of conjugacy classes
zbMATH Keywords: closed subgroups
zbMATH Keywords: totally categorical structures
reviewed by: H. D. Macpherson
MaRDI profile type: Publication
OpenAlex ID: W2072316428
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DOI: 10.1006/JABR.2001.9110
description: scientific article; zbMATH DE number 1763559