What Are the Properties of Dihedral Groups?

[Greg Bernhardt]

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
$\theta_k = \frac{2\pi k}{n}$

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[fresh_42]

Dihedral groups must not be confused with Dicyclic groups. Here is an interesting graphic how they are related: https://en.wikipedia.org/wiki/Dicyclic_group#Properties

see also: https://www.physicsforums.com/threads/what-is-a-dicyclic-group.762865/