Prove that for all fixed $\displaystyle k\in\mathbb{Q}$ such that k isn't 0 the map $\displaystyle \varphi :\mathbb{Q}\to\mathbb{Q}$ defined by $\displaystyle \varphi (q)=kq$ is an automorphism of $\displaystyle \mathbb{Q}$.

So I need to show it is an isomorphism. I am having a problem with the homomorphism part though.

Let $\displaystyle q,r\in\mathbb{Q}$. $\displaystyle \varphi (qr)=kqr$ but $\displaystyle \varphi (q)\varphi(r)=kqkr=k^2qr$.

What do I need to do?