Show there are infinitely many automorphisms of the group $\mathbb{Q}^*$.
I am not sure how show this. If we were dealing with ring automorphisms $\varphi:\mathbb{Q} \to \mathbb{Q}$, then the fact that $\varphi(1)=1$ makes such a ring automorphism unique. However, how can we show that with groups that there are inifinitely many such automorphisms.