Simplicity of some automorphism groups.
type: Item
instance of: scholarly article
title: Simplicity of some automorphism groups.
zbMATH Open document ID: 1244.20002
author: Dugald Macpherson
author: Katrin Tent
published in: Journal of Algebra
publication date: 2012-02-10T00:00:00Z
review text: This article deals with the automorphism groups of homogeneous (relational) structures \(M\). These were in various special cases known to be simple; this is no general truth but as the authors point out in the introduction ``if \(\Aut(M)\) is not simple, then it has some \textit{obvious} proper non-trivial normal subgroups, explicable in terms of its actions''. The article provides for the first time a general result explaining the tendency to simplicity such groups have. It is shown (Theorem 1.1) that given a \textit{free} homogeneous structure \(M\) (meaning \(M\) is the Fraïssé limit of a class with a stronger form of amalgamation property), then provided \(\Aut(M)\) is a transitive, proper subgroup of \(M\), it is simple. In particular this result encompasses the (previously known) cases of: the random graph [\textit{J. K. Truss}, Math. Proc. Camb. Philos. Soc. 105, No. 2, 223-236 (1989; Zbl 0687.20003)], the random \(K_n\)-free graph [\textit{M. Rubin}, unpublished], and the random tournament [\textit{E. Jaligot}, unpublished]. The methods are borrowed both from descriptive set theory (the automorphism group is a Polish group) and model theory, with an abstract analogue of stability-theoretic independence in the context of Fraïssé limits (though no stability theory is used in the article). The authors ``believe [their] method to have considerable potential for further generalization''.
reviewed by: Adrien Deloro
Mathematics Subject Classification ID: 20B27
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 20A15
zbMATH DE Number: 6005095
zbMATH Keywords: simple permutation groups
zbMATH Keywords: automorphism groups
zbMATH Keywords: Fraïssé limits
zbMATH Keywords: Polish groups
MaRDI profile type: Publication
Wikidata QID: Q61848359
full work available at URL: j.jalgebra.2011.05.021
OpenAlex ID: W2024767348
cites work: Q3995219
cites work: A census of infinite distance-transitive graphs
cites work: Some isometry groups of the Urysohn space
cites work: The classification of countable homogeneous directed graphs and countable homogeneous 𝑛-tournaments
cites work: Structure of partially ordered sets with transitive automorphism groups
cites work: Automorphism Groups of Infinite Semilinear Orders (I)
cites work: Q5819562
cites work: Infinite Dimensional Classical Groups
cites work: Automorphism groups of countable highly homogeneous partially ordered sets
cites work: Countable homogeneous relational structures and ℵ0-categorical theories
cites work: Extending partial isomorphisms for the small index property of many \(\omega\)-categorical structures
cites work: Intersection matrices for finite permutation groups
cites work: Q4003410
cites work: A new strongly minimal set
cites work: Automorphisms of homogeneous structures
cites work: Q4323294
cites work: Turbulence, amalgamation, and generic automorphisms of homogeneous structures
cites work: Les automorphismes d'un ensemble fortement minimal
cites work: A survey of homogeneous structures
cites work: Q3341041
cites work: Q4396494
cites work: Free polygons, twin trees, and \(\mathrm{CAT}(1)\)-spaces
cites work: The group of almost automorphisms of the countable universal graph
cites work: Q3973777
cites work: The automorphism group of the random graph: four conjugates good, three conjugates better.
cites work: Q4309356
DOI: 10.1016/J.JALGEBRA.2011.05.021
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description: scientific article; zbMATH DE number 6005095