Let $G=(\Bbb Q\setminus\{{0\}},∗)$ be a group of the rational numbers under multiplication. How to find the structure of ${\rm Aut}(G)$?
I show that if $f$ is a automorphism then it maps a basis of $G$ to another basis.
Conversely a automorphism can be constructed given a mapping between two basis.
Obviously the set of prime numbers is a basis, so I want to find all basis of this group. But such basis are not just prime numbers (for example, you can make $x_n$ the product of the first n prime numbers, so that the elements also form a basis).