How do I prove isomorphism?

Question

I need to prove this: $$S_{\mathbb{N}}\cong S_{\mathbb{Z}}$$ ($S$ means permutation).

I'd like to get ideas how to prove it...

Thank you!

Answer 1

Essentially, you wish to construct an isomorphism $\alpha : S_\mathbb{N} \to S_\mathbb{Z}$.

But what are the elements of $S_\mathbb{N}$? They're just $f : \mathbb{N} \to \mathbb{N}$ bijective. Likewise for $\mathbb{Z}$.

Hint: there is some $g : \mathbb{N} \to \mathbb{Z}$ that is bijective. Can you use this to construct an explicit isomorphism? What is a natural thing to do to some $f : \mathbb{N} \to \mathbb{N}$?