Proving Z_qp = Z_q × Z_p for distinct primes q and p

[Luris]

Hi, I'm going through a group theory text on my own, as it is not formally covered in my undergrad curriculum. I've had a good (multiple course) background in the basics, differential equations, linear algebra with Hilbert spaces, etc., in my undergrad coursework in my physics major. I'd like to use this thread in asking some questions in group theory, as I'm learning it from the textbook.

Here is my first question:

Prove: If q and p are prime then the cyclic group of order q*p can be factorized into the direct product of cyclic groups of order q and order p

(Symbolically) Prove: $$Z_{qp} = Z_q \times Z_p$$

Since this problem was introduced right after they established the representation where any element in the cyclic group can be represented by a power of $$a = exp(2i\pi/N)$$, where N is the order of the group, I tried expressing $$Z_n = exp(2i\pi/n)$$ and then verifying the two sides... and I'm not getting an equality. I'm wondering how to prove it?

 

[morphism]

I think the ingredient you're missing is the Chinese remainder theorem. (http://planetmath.org/encyclopedia/ChineseRemainderTheoremProof.html )

By the way, there is a important subtlety you should be aware of: $Z_{qp} = Z_q \times Z_p$ is not an actual equality - it's an isomorphism.