You can cop out a little bit by noting that your map is between finite sets. Since the map is injective, it is automatically surjective.
I am trying to show that is an ismorphism,
defined by where
where U(n) is the multiplicative group of units (refresher: numbers relativly prime to n)
and is the external direct product.
so clearly is well defined
I have shown it is 1-1
So my problem is...
I am having trouble showing is it onto (to show to that it is a bijection)
so let . So b is an arbitrary element of .
so what we need to do is find some such that
(we must find a x that maps to b... defn of onto, right?)
Keep in mind that we know that since
So I thought maybe ?
then by the definition of alpha,
Now we just need to get rid of the t in the first component and s in the second component to map to that arbitrary element
Yikes sorry for all the description. All help would be much appriciated, this actually stumped our class for a few minutes at the end of class today!
Thanks! in advance!