(sighs) I need someone to confirm if I'm going insane again.

If it's an automorphism then it's a homomorphism. But:Prove that for each fixed nonzero $\displaystyle k \in \mathbb{Q}$ the map from $\displaystyle \mathbb{Q}$ to itself by $\displaystyle q \to kq$ is an automorphism of $\displaystyle \mathbb{Q}$.

$\displaystyle \phi _k : \mathbb{Q} \to \mathbb{Q}: q \mapsto kq$ is not a homomorphism, except for k = 1:

Let $\displaystyle f,g \in \mathbb{Q}$.

$\displaystyle \phi _k (f g ) = kfg$

and

$\displaystyle \phi _k (f) \phi_k(g) = (kf) (kg) = k^2fg$

What am I missing this time? (It's just been that kind of day.)

-Dan