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What is a dicyclic group

  1. Jul 23, 2014 #1
    Definition/Summary

    The dicyclic group or generalized quaternion group Dic(n) is a nonabelian group with order 4n that is related to the cyclic group Z(2n).

    It is closely related to the dihedral group.

    Equations

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

    Its elements are
    [itex]Dic_n = \{a^k, ba^k : 0 \leq k < 2n \}[/itex]

    Its "reflection-like" elements all have order 4, unlike the similar elements of the dihedral group with order 2.
    [itex](ba^k)^4 = 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} i \cos\theta_k & - i \sin\theta_k \\ - i \sin\theta_k & - i \cos\theta_k \end{pmatrix}[/itex]
    where
    [itex]\theta_k = \frac{\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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