Group of recursive permutations

Origin
The group of recursive permutations was first studied by Clement F. Kent in his paper Constructive Analogues of the Group of Permutations of the Natural Numbers.

Definition
Let $$\N$$ denote the set of natural numbers. The group of recursive permutations of $$N$$ is the group of all those permutations of $$\N$$ that arise via recursive functions.

This group contains the finitary symmetric group on the natural numbers, and is contained inside the symmetric group on natural numbers (the group of all permutations on the natural numbers).