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dicyclic group

 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 $a^{2n} = e ,\ b^2 = a^n ,\ bab^{-1} = a^{-1}$ Its elements are $Dic_n = \{a^k, ba^k : 0 \leq k < 2n \}$ Its "reflection-like" elements all have order 4, unlike the similar elements of the dihedral group with order 2. $(ba^k)^4 = e$

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 Breakdown Mathematics > Algebra >> Group Theory

 Extended explanation This group may be realized as the matrices $a^k = \begin{pmatrix} \cos\theta_k & - \sin\theta_k \\ \sin\theta_k & \cos\theta_k \end{pmatrix}$ $ba^k = \begin{pmatrix} i \cos\theta_k & - i \sin\theta_k \\ - i \sin\theta_k & - i \cos\theta_k \end{pmatrix}$ where $\theta_k = \frac{\pi k}{n}$