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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
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}
* 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!
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
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}
* 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!