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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.

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    It might help to start by figuring out the abstract structure of $\mathbb{Q}^{\ast}$.2012-02-06

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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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    @RickyDemer: You are right. By the way $\mathbb Q$ also has uncountably many subrings, for similar reasons.2012-02-06