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HOMOGENEOUS STRUCTURES WITH NONUNIVERSAL AUTOMORPHISM GROUPS

Part of: Foundations Model theory

Published online by Cambridge University Press:  22 June 2020

WIESŁAW KUBIŚ  and
SAHARON SHELAH
WIESŁAW KUBIŚ
Affiliation:
INSTITUTE OF MATHEMATICS CZECH ACADEMY OF SCIENCES CZECH REPUBLIC and CARDINAL STEFAN WYSZYŃSKI UNIVERSITYWARSAW, POLANDE-mail: kubis@math.cas.cz
SAHARON SHELAH
Affiliation:
EINSTEIN INSTITUTE OF MATHEMATICS HEBREW UNIVERSITY OF JERUSALEMISRAEL and DEPARTMENT OF MATHEMATICS RUTGERS UNIVERSITYNEW BRUNSWICK, NJ, USAE-mail: shelah@math.huji.ac.il
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Abstract

We present three examples of countable homogeneous structures (also called Fraïssé limits) whose automorphism groups are not universal, namely, fail to contain isomorphic copies of all automorphism groups of their substructures.

Our first example is a particular case of a rather general construction on Fraïssé classes, which we call diversification, leading to automorphism groups containing copies of all finite groups. Our second example is a special case of another general construction on Fraïssé classes, the mixed sums, leading to a Fraïssé class with all finite symmetric groups appearing as automorphism groups and at the same time with a torsion-free automorphism group of its Fraïssé limit. Our last example is a Fraïssé class of finite models with arbitrarily large finite abelian automorphism groups, such that the automorphism group of its Fraïssé limit is again torsion-free.

Keywords

universal automorphism groupFraïssé limitFraïssé classdiversificationmixed sum

MSC classification

Primary: 20A15: Applications of logic to group theory
Secondary: 03C15: Denumerable structures 03C50: Models with special properties (saturated, rigid, etc.)
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 2 , June 2020 , pp. 817 - 827
Copyright
© The Association for Symbolic Logic 2020

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References

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