Abelian subgroups of order automorphism group $({\rm Aut}(\mathbb R,\le), \circ )$

Asked Jan 31 '24 at 20:09

Active Jan 31 '24 at 20:40

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I am searching for any results regarding Abelian subgroups of $({\rm Aut}(\mathbb R,\le), \circ )$, the order automorphism group of $\mathbb R$ (order automorphisms of $\mathbb R$ with the composition operation).

Are there any known results on the subject? Is there any book that covers the particular subject or any relevant monograph?

edited Jan 31 '24 at 20:40

Shaun

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asked Jan 31 '24 at 20:09

Crispost

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In Peter Neumann's lecture notes on infinite permutation groups https://arxiv.org/abs/2307.11564 there are some results on automorphisms of the rational line. Some of these will carry over to the real line. – David A. Craven Jan 31 '24 at 21:27
Can you clarify what you mean by $\mathrm{Aut}(\mathbb{R}, \leq)$? If that means the ring [i.e. field] automorphisms of $\mathbb{R}$ that are order preserving, then the automorphism group is trivial. See here. Do you instead mean that $(\mathbb{R}, \leq)$ is being considered only as an ordered set? – David M Feb 16 '24 at 06:59
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Does this answer your question? Is an automorphism of the field of real numbers the identity map? – Ricky Feb 20 '24 at 00:40

Abelian subgroups of order automorphism group $({\rm Aut}(\mathbb R,\le), \circ )$

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