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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 $\mathbb{Q}^{\ast}$. –  Qiaochu Yuan Feb 6 '12 at 6:51
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up vote 8 down vote accepted

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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I think this would also show that there are uncountably many automorphisms of $\langle \mathbb{Q}^*,\cdot \rangle$. $\hspace{1 in}$ –  Ricky Demer Feb 6 '12 at 6:55
There has to be simpler way of looking at this problem. I haven't seen the presented direct sum before. How do you get it? –  Galois Feb 6 '12 at 7:21
Think of prime decomposition! The first factor is the sign $\pm$. –  Bruno Joyal Feb 6 '12 at 7:24
@Galois, that's the fundamental theorem of arithmetic applied to $\mathbb Q$, that is, to the numerator and denominator of a fraction. –  lhf Feb 6 '12 at 9:38
@RickyDemer: You are right. By the way $\mathbb Q$ also has uncountably many subrings, for similar reasons. –  Marc van Leeuwen Feb 6 '12 at 12:31
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