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Homework Help: Group Theroy problem

  1. Apr 17, 2010 #1
    1. The problem statement, all variables and given/known data
    In Z, let H=<5> and K=<7>. Prove that Z=HK. Is Z the internal direct product of <5> and <7>?

    2. Relevant equations

    3. The attempt at a solution
    Since 3(5)-2(7)=1, every integer n element of Z can be written as 3n(5)+(-2n)(7), and n is an element of HK( where group operation is addition and powers written as multiples). Therefore Z=HK. Since the intersection of H and K =<35>, then Z is not equal to H direct product K.
    I think this is right but I am not sure on some of the justification of my claims. Can any one help me with the justification?
  2. jcsd
  3. Apr 17, 2010 #2


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    Homework Helper

    What part are you finding a hard time justifying? You've shown Z=HK and 5^2*7^2=35*35 so it's not a direct product. What else do you need?
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