[abstract algebra] Isomorphic group of units

[nonequilibrium]

Homework Statement

Given that gcd(n,m)=1, prove that $\mathbb Z_{nm}^\times = \mathbb Z_n^\times \oplus \mathbb Z_m^\times$.

Homework Equations

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The Attempt at a Solution

I can prove both groups have the same amount of elements (using Euler's totient function), but I can't figure out how to prove the isomorphism. One way would be to construct the isomorphism, but I can't seem to find one.

 

[Barre]

Perhaps consider function $f: \mathbb Z_{nm}^\times \rightarrow \mathbb Z_n^\times \oplus \mathbb Z_m^\times$ such that f(a) = (a mod n, a mod m). Since if $a \in \mathbb Z_{nm}^\times$ then a is relatively prime to mn, so there is an integer solution x,y to the equations $ax +mny =1$. Taking the equation, you can re-write it to $mny = 1 - ax$, which means $ax$ is congruent 1 modulo m, and congruent 1 modulo n. So actually a has an inverse mod m and mod n (in this case x). So f maps units to units. You can prove surjectiveness by Chinese Remainder Theorem, and since sets are finite this would imply bijectivity.

/edit reworded slightly.

 

[nonequilibrium]

Thank you!

 

[Hurkyl]

As a reminder to all homework helpers, we're supposed to help the student solve a problem, not to do it for them. Barre's post is a demonstration of exactly what not to do. Unfortunately, it's too late and the original poster has already received the complete solution to his homework problem.