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Group Theoretic Properties of the Group of Computable Automorphisms of a Countable Dense Linear Order

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Abstract

We compare Aut(Q), the classical automorphism group of a countable dense linear order, with Aut c (Q), the group of all computable automorphisms of such an order. They have a number of similarities, including the facts that every element of each group is a commutator and each group has exactly three nontrivial normal subgroups. However, the standard proofs of these facts in Aut(Q) do not work for Aut c (Q). Also, Aut(Q) has three fundamental properties which fail in Aut c (Q): it is divisible, every element is a commutator of itself with some other element, and two elements are conjugate if and only if they have isomorphic orbital structures.

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Lempp, S., McCoy, C., Morozov, A. et al. Group Theoretic Properties of the Group of Computable Automorphisms of a Countable Dense Linear Order. Order 19, 343–364 (2002). https://doi.org/10.1023/A:1022878003199

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  • DOI: https://doi.org/10.1023/A:1022878003199

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