Prove that structure $(Q/{0}; =;*)$ has infinite number of automorphisms

Question

Prove that structure $(Q/{0}; =;*)$ has infinite number of automorphisms I think it's about we don't have $<$ or $>$, so we don't have strict order in structure, so we can do whatever we want? but i dont know how to say it more formal/

Answer 1

Hint: Take any permutation (i.e. bijection to itself) of the set of the prime numbers $P$: $\phi:P\to P$. Now map every rational number $q=\pm p_1^{\lambda_1}\cdots p_n^{\lambda_n}$, where $\lambda_1,\ldots\lambda_n\in\mathbb Z$ into $\pm \phi(p_1)^{\lambda_1}\cdots \phi(p_n)^{\lambda_n}$. Prove that this map is an automorphism.