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Klein Four Group's action

  1. Oct 16, 2014 #1

    ChrisVer

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    Can someone please read the attached file and tell me his ideas? I want to be sure I understand the action of the Klein Four group....Is my interpretation correct?

    I have some problem though. The Klein Four group has 4 elements, and it is able to describe the reflective symmetries of a square (as I interpreted it).
    The dihedral group however, $D_2$ for example should be the same group right? (if I see the one $Z_2$ of the KF group as a cyclic group and the other $Z_2$ are reflection). $D_2$ however does not describe a square....
     

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  3. Oct 19, 2014 #2
    That should be easy to work out from the definition of dihedral group D(n): generators a and b with
    an = b2 = e
    a.b = b.a-1

    Note what happens when n = 2. What is a-1 in that case?
     
  4. Oct 19, 2014 #3

    ChrisVer

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    Hi,

    How is it connected to the dihedral group?

    Because the D(2) is of order 4, and since it's not cyclic C(4) it has to be (at least isomorphic to) the Klein Four group?

    [itex]D_2 = (e , a, b, ab) [/itex]
    and [itex]a^2=e \Rightarrow a^{-1}=a[/itex]
    So I wonder for n=2, there is no distinction between rotations and reflections (rotations are done by 180o)
     
  5. Oct 19, 2014 #4
    I'll work out D2 or Dih(2). It has
    a2 = b2 = e
    a.b = b.a-1
    But from the definition of a, a-1 = a, and thus, a.b = b.a. Thus, all the group's elements commute with each other.

    The n = 2 rotation and reflection elements:
    • Identity
    • 180d rotation
    • Two reflections
    They all commute with each other.

    Rotation: $$ \begin{pmatrix} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} $$
    Reflection: $$ \begin{pmatrix} \cos \theta & \sin \theta \\ \sin \theta & - \cos \theta \end{pmatrix} $$
    where θ is the rotation angle. For D2, it is 0 and 180d. In general, ## \theta_{rot}(k) = 2\pi k / n ## and
    ## \theta_{refl}(k) = \theta_0 + 2\pi k / n ## for k = 0, 1, ..., n-1, where θ0 is arbitrary.
     
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