Consider the abelian group $(\mathbb{Q}_{>0}, \times)$. What automorphisms exist for this group? I can only think of the trivial one and of $\phi(q) = \frac{1}{q}$.
If we relax the problem to injective homomophisms from $(\mathbb{Q}_{>0}, \times)$ to itself, do we get additional results?
Hint $(\Bbb Q_{> 0}, \cdot)$ is isomorphic to $(\Bbb Z, +) \oplus (\Bbb Z, +) \oplus \cdots$.
It's easier to first talk about endomorphisms because these have a ring structure. In fact, as has been mentioned a few times already, this group is a countable direct sum $\bigoplus_p \mathbb{Z}$ of a copy of $\mathbb{Z}$ for every prime, and hence in some respects it behaves like a vector space. In particular, its endomorphism ring $\text{End}(\bigoplus_p \mathbb{Z})$ is a ring of matrices: more precisely, it's the ring of integer matrices with countably many rows and columns, but where there are a finite number of entries in each column (column-finite matrices for short). This condition ensures that multiplying such a matrix by a "vector" in $\bigoplus_p \mathbb{Z}$ is well-defined.
The automorphism group is then the group of units of this ring: that is, it's the group of invertible column-finite matrices over $\mathbb{Z}$. I think that is as simple of a description as you're going to get. This is a very large group; it includes the groups $\text{GL}_n(\mathbb{Z})$ for all $n$ as proper subgroups, as well as the group $\text{Aut}(\mathbb{N})$ of all permutations of a countable set.
