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?
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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 $\alpha(x)\ne\gamma(x)$.
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