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For an odd integer $n$, find an explicit isomorphism between $\mathbb{Z}^{\times}_n$ and $\mathbb{Z}^{\times}_{2n}$.
How do I do this? I don't really know where to start. I can easily find bijections but am yet to find a structure preserving one.

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    are you familiar with the Chinese Remainder Theorem?2011-05-19
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    @Qiaochu: How can u find a bijection when their orders is different.2011-05-19
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    @Chandru: Their orders are not different. $\# (\mathbb{Z}/n \mathbb{Z})^{\times} = \phi(n)$ and $\phi(n)=\phi(2n)$ if $2 \nmid n$.2011-05-19
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    @Brandon: Yeah this is what I wanted to know.2011-05-19

3 Answers 3