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dihedral group

 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 $a^n = b^2 = e ,\ bab^{-1} = a^{-1}$ Its elements are $D_n = \{a^k, ba^k : 0 \leq k < n \}$ All the "reflection" elements have order 2: $(ba^k)^2 = e$

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 Breakdown Mathematics > Algebra >> Group Theory

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