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dihedral group
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Definition/Summary
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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). |
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Equations
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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] |
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Recent forum threads on dihedral group
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Breakdown
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Mathematics
> Algebra
>> Group Theory
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Extended explanation
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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] |
Commentary
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