Group of computable permutations

Question

Why group of computable permutations of natural numbers is not finitely generated?

It is obvious for all permutations but why it is also true for computable permutations?

Answer 1

If $\{\pi_1,\pi_2,\dots,\pi_n\}$ is a finite set of computable permutations of $\mathbb N$, then the group $G=\langle\pi_1,\pi_2,\dots,\pi_n\rangle$ that they generate is recursively enumerable. Then it's a straightforward diagonalization to construct a computable permutation $\alpha$ (an involution, say) which is not in $G$. At each stage of the construction, you look at the first natural $x$ for which $\alpha(x)$ has not yet been defined, and the first permutation $\gamma\in G$ which has not yet been "spoiled", and you extend $\alpha$ so that $\alpha(x)\ne\gamma(x)$.