Geodesic
The categoricity of the group of all computable automorphisms of the rational numbers
Algebra i logika, Tome 46 (2007) no. 5, pp. 649-662.

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We prove that there is a first-order sentence $\varphi$ such that the group of all computable automorphisms of the ordering of the rational numbers is its only model among the groups that are embeddable in the group of all computable permutations.
Keywords: group of all computable automorphisms of rational numbers, finitely axiomatizable theory, categorical theory. 
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A. S. Morozov; J. K. Truss. The categoricity of the group of all computable automorphisms of the rational numbers. Algebra i logika, Tome 46 (2007) no. 5, pp. 649-662. http://geodesic.mathdoc.fr/item/AL_2007_46_5_a6/

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