Can we define a binary operation on $\mathbb Z$ to make it a vector space over $\mathbb Q$?

Question

It is known that any infinite cyclic group , in particular $(\mathbb Z, +)$ , can never be a vector space . So we may ask , Can we define an operation $*$ on $\mathbb Z$ such that $(\mathbb Z , *)$ becomes an abelian non-cyclic group such that it can be a vector space over some field with suitable scalar operation ? In particular can we make $(\mathbb Z , *)$ a vector space over $\mathbb Q$ ?

Answer 1

Here is what Git Gud is saying: If you remove the $+$ operation from $\mathbb{Z}$, the only information you are left with is that you have a countable set. So this is equivalent to asking if their are countable vector spaces, and any $\mathbb{Q}^n$ works.

More concretely, fix your favorite bijection $f: \mathbb{Z} \to \mathbb{Q}$ that fixes $0$. You can force this to be an isomorphism of vector spaces by defining

$$a \ast b := f^{-1} \Big( f(a) + f(b)\Big)$$

and

$$qa = f^{-1}\Big(qf(a)\Big)$$