Prove that .
I am not seeing it.
If we can write where and are the primes in increasing order.
Thus, I think the natural map is .
Of course, the infinite product is really finite and the infinite sum is really finite because
almost all terms are 1 in the infinite product and almost all terms are zero in the infinite sum.
Now of course since addition of exponents is adding the exponents.
This happens to be one-to-one and onto.
Argh! I spent the whole night thinking of the problem, and now that I have found a solution, I see that ThePerfectHacker and NonCommAlg have got here before me and deprived me of the chance to show off.
Only kidding. I come here to help, to learn from others and to share what I know, not to show off.
Anyway, this is a very beautiful problem. The fact that every positive rational number can be written uniquely in the form where is very beautiful indeed.
Yeah, as stated in my sheet of group theory questions it is .
It must be incorrect as stated because I am pretty sure .
The first has two elements of finite order while the second has only one of finite order 1.
And we have demonstrated the isomorphism to the latter.