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[abstract algebra] Isomorphic group of units

  1. Aug 26, 2011 #1
    1. The problem statement, all variables and given/known data
    Given that gcd(n,m)=1, prove that [itex]\mathbb Z_{nm}^\times = \mathbb Z_n^\times \oplus \mathbb Z_m^\times[/itex].

    2. Relevant equations
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    3. 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.
     
  2. jcsd
  3. Aug 26, 2011 #2
    Perhaps concider function [itex] f: \mathbb Z_{nm}^\times \rightarrow \mathbb Z_n^\times \oplus \mathbb Z_m^\times [/itex] such that f(a) = (a mod n, a mod m). Since if [itex]a \in \mathbb Z_{nm}^\times[/itex] then a is relatively prime to mn, so there is an integer solution x,y to the equations [itex]ax +mny =1[/itex]. Taking the equation, you can re-write it to [itex]mny = 1 - ax[/itex], which means [itex]ax[/itex] 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.
     
    Last edited: Aug 27, 2011
  4. Aug 27, 2011 #3
    Thank you!
     
  5. Aug 27, 2011 #4

    Hurkyl

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    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. :frown:
     
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