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What is a dihedral group

  1. Jul 23, 2014 #1
    Definition/Summary

    The dihedral group D(n) / Dih(n) is a nonabelian group with order 2n that is related to the cyclic group Z(n).

    The group of symmetries of a regular n-sided polygon under rotation and reflection is a realization of it. The pure rotations form the cyclic group Z(n), while the reflections form its coset in D(n). The quotient group is Z(2).

    Equations

    It has two generators, a and b, which satisfy
    [itex]a^n = b^2 = e ,\ bab^{-1} = a^{-1}[/itex]

    Its elements are
    [itex]D_n = \{a^k, ba^k : 0 \leq k < n \}[/itex]

    All the "reflection" elements have order 2:
    [itex](ba^k)^2 = e[/itex]

    Extended explanation

    This group may be realized as the matrices
    [itex]a^k = \begin{pmatrix} \cos\theta_k & - \sin\theta_k \\ \sin\theta_k & \cos\theta_k \end{pmatrix}[/itex]
    [itex]ba^k = \begin{pmatrix} \cos\theta_k & - \sin\theta_k \\ - \sin\theta_k & - \cos\theta_k \end{pmatrix}[/itex]
    where
    [itex]\theta_k = \frac{2\pi k}{n}[/itex]

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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