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Prove direct product of G1 x G2 is abelian

  1. Oct 31, 2010 #1
    1. The problem statement, all variables and given/known data

    Prove that if G1 and G2 are abelian, then the direct product G1 x G2 is abelian.


    2. Relevant equations



    3. The attempt at a solution
    let G1 and G2 be abelian. Then for a1,a2,b1,b2, we have a1b1=b1a1 and a2b2=b2a2.
    The direct product is the set of all ordered pairs (x1,x2) such that x1 is in G1 and x2 is in G2.
    Let x1 be in G1 and x2 be in G2.
    Then G1 x G= (x1, x2)
    I'm not quite sure how to show this is abelian.
     
  2. jcsd
  3. Oct 31, 2010 #2
    Take two elements in G1 x G2. What does these elements look like? What happens if you multiply them?
     
  4. Oct 31, 2010 #3
    Let a be in G1 b in G2. Then ab=ba
     
  5. Oct 31, 2010 #4
    No, I take an element in G_1 x G_2. What does this look like?
     
  6. Oct 31, 2010 #5
    oh, so we have (a, b) and (c,d)
    We have (a,b)(c,d)=(ac,bd)
    But ac=ca and bd=db
    So (ac,bd)=(ca,db)=(c,d)(a,b)
     
  7. Oct 31, 2010 #6
    Yes! good job!
     
  8. Oct 31, 2010 #7
    That's really all I have to do? it seems so simple.
     
  9. Oct 31, 2010 #8
    Yes, it's that simple :smile:
     
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