Finite covers with finite kernels
type: Item
instance of: scholarly article
title: Finite covers with finite kernels
zbMATH Open document ID: 0892.03011
DOI: 10.1016/S0168-0072(97)00018-3
author: David M. Evans
published in: Annals of Pure and Applied Logic
publication date: 1998-06-30T00:00:00Z
review text: The author considers certain infinite imprimitive automorphism groups, motivated by model-theoretic questions on classes of \(\aleph_0\)-categorical structures. The basic notion is that of a permutation structure, that is a set \(X\) together with a closed subgroup of \(\text{Sym}(X)\) (denoted \(\text{Aut}(X)\)). If \(C\), \(W\) are permutation structures, then a finite cover is a surjection \(\pi:C\rightarrow W\) such that the \(\{\pi^{-1}(w)\}\) (where \(w \in W\)) are finite and form an \(\text{Aut}(C)\)-invariant partition of \(C\), and the induced map \(\rho: \text{Aut}(C) \rightarrow \text{Aut}(W)\) given by \(\rho g(w)=\pi(g\pi^{-1}(w))\) has image \(\text{Aut}(W)\). The kernel of the cover is \(\text{ker}(\mu)\). The emphasis is on the case when \(W\) is \(\aleph_0\)-categorical (so \(\text{Aut}(W)\) has finitely many orbits on \(k\)-sets for all finite \(k\)) and \(\text{ker}(\mu)\) is finite. The author examines possible kernels, and possible covers with given kernel, under certain hypotheses. In particular, he conjectures that under natural hypotheses (on \(W\) and the class of covers), for a given permutation structure \(W\) there is a natural number \(r\) such that the kernel of any finite cover is at most \(r\)-generated. This has since been verified, by the author [J. Algebra 193, 214-238 (1997; Zbl 0884.03041)] and also by J. Koshan and by E. Hrushovski. The author introduces natural conditions -- untwisted, locally trivial, locally transitive (sometimes relativised to an orbit) -- and gives reduction lemmas to these situations. He develops a theory of finite coverings of digraphs, and uses this, under certain hypotheses, to define a universal covering of a structure, and also to define an analogue of the first homology group, which has all sufficiently nice kernels as quotients. This theory works well if there is a strong type, a privileged 1-type which exists in many \(\omega\)-categorical structures. In the final section, the author gives a reduc…
Mathematics Subject Classification ID: 03C35
Mathematics Subject Classification ID: 20B27
zbMATH DE Number: 1109921
zbMATH Keywords: finite cover
zbMATH Keywords: aleph-zero categorical structures
zbMATH Keywords: automorphism groups
zbMATH Keywords: coverings of digraphs
zbMATH Keywords: permutation structure
zbMATH Keywords: untwisted
zbMATH Keywords: locally trivial
zbMATH Keywords: locally transitive
zbMATH Keywords: analogue of the first homology group
zbMATH Keywords: kernels
Wikidata QID: Q56269725
reviewed by: H. D. Macpherson
MaRDI profile type: Publication
cites work: Invariant subgroups of \(^ V V\)
cites work: What's so special about \((\mathbb{Z}/4\mathbb{Z})^ \omega\)?
cites work: On one-based theories
cites work: Splitting of Finite Covers of ℵ0 -Categorical Structures
cites work: On the automorphism groups of finite covers
cites work: Q3027108
cites work: Q5557005
cites work: Q4003410
cites work: Q4309354
cites work: Cohomology of Structures and Some Problems of Ahlbrandt and Ziegler
cites work: Endliche Gruppen I
cites work: Q4309359
DBLP publication ID: journals/apal/Evans97
description: scientific article; zbMATH DE number 1109921