TITLE: Automorphism of $\mathbb{Q}^*$
QUESTION [6 upvotes]: 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.

REPLY [9 votes]: Hint: $$\mathbb{Q}^\times \cong (\mathbb{Z}/ 2\mathbb{Z}) \oplus \bigoplus_{p}\mathbb{Z}$$
where the direct sum is indexed over the primes.
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