Can we find a binary operation s.t. the map $\phi(n)=n+1$ becomes an isomorphism from $\langle\Bbb Z,*\rangle$ onto $\langle\Bbb Z,.\rangle$?

Question

Can we find a suitable binary operation $*$ on $\mathbb{Z}$ such that the map $\phi : \mathbb{Z} \to \mathbb{Z}$ defined as $\phi(n)= n+1$ becomes an isomorphism from $\langle \mathbb{Z}, *\rangle $ onto $\langle \mathbb{Z}, .\rangle$ ? Here $.$ is the Usual multiplication.

This is a lot easier if the group $\langle \mathbb{Z}, +\rangle $ is considered instead of $\langle \mathbb{Z}, .\rangle$. Because then $a*b = a+b+1$ works just fine. But how can I define the operation here?

Edit : I think I got the hang of the answer from Arthur's comment.

Let, $a*b = (a+1)(b+1) - 1$.

Then, $\phi(a*b) = (a+1)(b+1) = \phi(a) \cdot \phi(b)$. I think this is correct.

Can anyone give me some hint?

Answer 1

Hint: Rephrase your $(\Bbb Z,+)$ solution from $a+b+1$ to $(a+1)+(b+1)-1$. Now see if you can't apply the same tactic.

Answer 2

If $\phi$ is an isomorphism then you know the inverse. The operation is the one induced by $(\mathbb{Z},+)$.