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.
Finding an explicit isomorphism between $\mathbb{Z}^{\times}_n$ and $\mathbb{Z}^{\times}_{2n}$
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$\begingroup$
abstract-algebra
finite-groups
abelian-groups
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3are you familiar with the Chinese Remainder Theorem? – 2011-05-19
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0@Qiaochu: How can u find a bijection when their orders is different. – 2011-05-19
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1@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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0@Brandon: Yeah this is what I wanted to know. – 2011-05-19