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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.

Clara
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Galois
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1 Answers1

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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.

Bruno Joyal
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    I think this would also show that there are uncountably many automorphisms of $\langle \mathbb{Q}^*,\cdot \rangle$. $\hspace{1 in}$ –  Feb 06 '12 at 06:55
  • 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 06 '12 at 07:21
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    Think of prime decomposition! The first factor is the sign $\pm$. – Bruno Joyal Feb 06 '12 at 07:24
  • @Galois, that's the fundamental theorem of arithmetic applied to $\mathbb Q$, that is, to the numerator and denominator of a fraction. – lhf Feb 06 '12 at 09:38
  • @RickyDemer: You are right. By the way $\mathbb Q$ also has uncountably many subrings, for similar reasons. – Marc van Leeuwen Feb 06 '12 at 12:31