Doesn't look like an automorphism to me...

Question

I am self-studying algebra using Dummit and Foote. One early exercise asks me to prove that for each fixed nonzero $k \in \mathbb{Q}$, the map $\varphi:\mathbb{Q} \to \mathbb{Q}$ defined as $\varphi(q) = kq$ is an automorphism. It's certainly a bijection, but I am having difficulty persuading myself it is a homomorphism. For let $p,q \in \mathbb{Q}$. Then \begin{align} \varphi(pq) & = kpq \end{align} while \begin{align} \varphi(p)\varphi(q) & = k^2pq, \end{align} and unless $k = 1$, these are not equal. Evidently I'm missing something obvious, but I can't figure out what.

Answer 1

I presume the question implicitly means for the semigroup operation to be addition, not multiplication.