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

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.