Direct product

  1. Let k be a positive integer.

    define G_k = {x| 1<= x <= k with gcd(x,k)=1}

    prove that:
    a)G_k is a group under multiplication modulos k (i can do that).

    b)G_nm = G_n x G_m be defining an isomorphism.
     
  2. jcsd
  3. matt grime

    matt grime 9,395
    Science Advisor
    Homework Helper

    What have you done for b)? There is only one possible way you can think of to write out a map from G_nm to G_n x G_m, so prove it is an isomorphism. Remember, G_n x G_m looks like pars (x,y)....
     
  4. We can use the Chinese Remainder Theorem on this one.

    Define the mapping,
    [tex]\phi: G_{nm}\mapsto G_n\times G_m[/tex]
    As,
    [tex]\phi(x) = (x\bmod{n} , x\bmod{m})[/tex]

    1)The homomorphism part is trivial.
    2)The bijection part is covered by Chinese Remainder Theorem.
     
  5. mathwonk

    mathwonk 9,814
    Science Advisor
    Homework Helper

    but the point is to prove that theorem.
     
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