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

Question

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?

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
Does this answer your question? Is an automorphism of the field of real numbers the identity map? — Ricky, Feb 20 '24 at 00:40

score 0 · Answer 1 · answered May 21 '24 at 21:33

I have located the following result which is relevant to the question, in "The theory of Lattice-Ordered Groups -- V. M. Kopytov, N. Ya. Medvedev, page 36, Corollary 1.".

"The group of order automorphisms of Archimedean o-group is isomorphic to some subgroup of the multiplicative group $ \mathbb R^+ $ of positive reals."

As $ \mathbb R $ is an Archimedean o-group (same as "linearly ordered" or "totally ordered"), the statement applies to the reals themselves.

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

1 Answers1