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Outer automorphism groups of ordered permutation groups

Outer automorphism groups of ordered permutation groups Abstract. An in®nite linearly ordered set SY is called doubly homogeneous if its automorphism group AS acts 2-transitively on it. We show that any group G arises as outer automorphism group G q OutAS of the automorphism group AS, for some doubly homogeneous chain SY . 2000 Mathematics Subject Classi®cation: 20F28, 06F15, 20B22. 1 Introduction An in®nite linearly ordered set (``chain'') SY is called doubly homogeneous, if its automorphism group, i.e. the group of all order-preserving permutations, AS AutSY acts 2-transitively on it. Chains SY of this type and their automorphism groups AS have been intensively studied. They have been used e.g. for the construction of in®nite simple torsion-free groups (Higman [8]) or, in the theory of lattice-ordered groups (l-groups), for embedding arbitrary l-groups into simple divisible l-groups (Holland [9]). The normal subgroup lattices of the groups AS have been determined in [1, 3]. Obviously, all linearly ordered ®elds are examples for such chains. For a variety of further results, see Glass [6]. Here, we will be concerned with outer automorphism groups OutAS AutASaInnAS of the automorphism groups AS for doubly homogeneous chains SY . In the literature, many authors have dealt with the problem of determining which http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Outer automorphism groups of ordered permutation groups

Forum Mathematicum , Volume 14 (4) – Apr 15, 2002

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References (10)

Publisher
de Gruyter
Copyright
Copyright © 2002 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2002.026
Publisher site
See Article on Publisher Site

Abstract

Abstract. An in®nite linearly ordered set SY is called doubly homogeneous if its automorphism group AS acts 2-transitively on it. We show that any group G arises as outer automorphism group G q OutAS of the automorphism group AS, for some doubly homogeneous chain SY . 2000 Mathematics Subject Classi®cation: 20F28, 06F15, 20B22. 1 Introduction An in®nite linearly ordered set (``chain'') SY is called doubly homogeneous, if its automorphism group, i.e. the group of all order-preserving permutations, AS AutSY acts 2-transitively on it. Chains SY of this type and their automorphism groups AS have been intensively studied. They have been used e.g. for the construction of in®nite simple torsion-free groups (Higman [8]) or, in the theory of lattice-ordered groups (l-groups), for embedding arbitrary l-groups into simple divisible l-groups (Holland [9]). The normal subgroup lattices of the groups AS have been determined in [1, 3]. Obviously, all linearly ordered ®elds are examples for such chains. For a variety of further results, see Glass [6]. Here, we will be concerned with outer automorphism groups OutAS AutASaInnAS of the automorphism groups AS for doubly homogeneous chains SY . In the literature, many authors have dealt with the problem of determining which

Journal

Forum Mathematicumde Gruyter

Published: Apr 15, 2002

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