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Homework Help: Unit circle

  1. Feb 3, 2008 #1
    [SOLVED] unit circle

    1. The problem statement, all variables and given/known data
    My book contains the following problem:

    Let U be the multiplication group [itex] \{z \in C : |z| = 1\} [/itex]

    1) Let z_0 be in U. Show that [itex] U z_0 = \{ z z_0 : z \in U \}[/itex] is a subgroup of U, and compute U mod U z_0.
    2) To what group is U/<-1> isomorphic to?

    2. Relevant equations

    3. The attempt at a solution
    I think 1) is so insanely trivial it is not worth asking. The answer is clearly the trivial group, right?

    My book says that the answer to 2) is U, but it seems it should be the half-circle or the reals mod 2 or something. Why is it U?[tex]\in\in[/tex]
    Last edited: Feb 3, 2008
  2. jcsd
  3. Feb 3, 2008 #2


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    U can also be represented as an additive group. It's R/(Z*2pi). U/<-1> is R/(Z*pi). Is there any real difference? The 'half circle' is isomorphic to the 'full circle'.
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