TITLE: Finding an explicit isomorphism between $\mathbb{Z}^{\times}_n$ and $\mathbb{Z}^{\times}_{2n}$
QUESTION [8 upvotes]: 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.

REPLY [6 votes]: HINT $\: $ It's very easy. The natural quotient map $\rm\ a+2\:n\ \mathbb Z\ \mapsto\ a+n\ \mathbb Z\ $ is surjective (so injective) on units, $\:$  since $\rm\:n\:$ odd $\:\Rightarrow\: $ one of $\rm\:a,\ a+n\:$ is odd, $\:$ so is coprime to $\rm\:2\:n\iff$  coprime to $\rm\:n\:.$