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Isomorphism from group to a product group

  1. Nov 13, 2011 #1
    1. The problem statement, all variables and given/known data

    Determine whether or not G is isomorphic to the product group HXK.

    G=ℂx
    H={unit circle}
    K={Positive real numbers}



    2. Relevant equations

    Let H and K be subroups of G, and let f:HXK→G be the multiplication map, defined by f(h,k)=hk. Its image is the set HK={hk: h in H, k in K}.

    f is an isomorphism from the product group HXK to G iff H intersect K is the identity,HK=G, and also H and K are normal subgroups of G.


    3. The attempt at a solution

    My actual question is how am I supposed to look at the unit circle?
    I know if I view it as all the points (a,b) on the unit circle then its intersection with K would be the empty set and thus G wouldn't be isomorphic to HXK.

    but I have a feeling that is too simple. Am I potentially supposed to view the unit circle as cos(θ)+i*sin(θ) where θ is in ℝ?
     
    Last edited: Nov 13, 2011
  2. jcsd
  3. Nov 13, 2011 #2

    I like Serena

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    Homework Helper

    Any complex number in ℂx can be represented as re and the reverse holds as well.
    So z → (r,θ) is a bijection.
    If you define the operations in HxK to be regular multiplication, respectively regular addition, you have preservation of the operation, since it literally is the same.
     
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