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Order-automorphism groups of Archimedean ordered groups

Part of: Special aspects of infinite or finite groups Ordered structures Structure and classification of infinite or finite groups Abelian groups

Published online by Cambridge University Press:  23 December 2024

Greg Marks Open the ORCID record for Greg Marks [Opens in a new window]
Greg Marks*
Affiliation:
Department of Mathematics and Statistics, St. Louis University, St. Louis, MO 63103, USA
*
e-mail: [email protected]
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Abstract

The group of order-preserving automorphisms of a finitely generated Archimedean ordered group of rank $2$ is either infinite cyclic or trivial according as the ratio in $\mathbb {R}$ of the generators of the subgroup is or is not quadratic over $\mathbb {Q}.$ In the case of an Archimedean ordered group of rank $2$ that is not finitely generated, the group of order-preserving automorphisms is free abelian. Criteria determining the rank of this free abelian group are established.

Keywords

Ordered groupArchimedeanorder-preserving automorphism

MSC classification

Primary: 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces 20E36: Automorphisms of infinite groups 20F60: Ordered groups
Secondary: 20K30: Automorphisms, homomorphisms, endomorphisms, etc.
Type
Article
Information
Canadian Mathematical Bulletin , Volume 68 , Issue 2 , June 2025 , pp. 628 - 633
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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