4.2: Dihedral Groups

We can think of finite cyclic groups as groups that describe rotational symmetry. In particular, \(R_n\) is the group of rotational symmetries of a regular \(n\)-gon. Dihedral groups are those groups that describe both rotational and reflectional symmetry of regular \(n\)-gons.

Definition: Dihedral Group

For \(n\geq 3\), the dihedral group \(D_n\) is defined to be the group consisting of the symmetry actions of a regular \(n\)-gon, where the operation is composition of actions.

For example, as we’ve seen, \(D_3\) and \(D_4\) are the symmetry groups of equilateral triangles and squares, respectively. The symmetry group of a regular pentagon is denoted by \(D_5\). It is a well-known fact from geometry that the composition of two reflections in the plane is a rotation by twice the angle between the reflecting lines.

Theorem \(\PageIndex{1}\)

The group \(D_n\) is a non-abelian group of order \(2n\).

Theorem \(\PageIndex{2}\): Generators \(D_n\)

Fix \(n\geq 3\) and consider \(D_n\). Let \(r\) be rotation clockwise by \(360^{\circ}/n\) and let \(s\) and \(s'\) be any two adjacent reflections of a regular \(n\)-gon. Then

1. \(D_n=\langle r,s\rangle =\{\underbrace{e,r,r^2,\ldots, r^{n-1}}_{\text{rotations}},\underbrace{s,sr,sr^2,\ldots,sr^{n-1}}_{\text{reflections}}\}\) and
2. \(D_n=\langle s,s'\rangle = \text{all possible products of }s\text{ and }s'\).

The next result is an obvious corollary of Theorem \(\PageIndex{2}\).

Corollary \(\PageIndex{1}\)

For \(n\geq 3\), \(R_n\leq D_n\).

The following theorem generalizes many of the relations we have witnessed in the Cayley diagrams for the dihedral groups \(D_3\) and \(D_4\).

Exercise \(\PageIndex{1}\)

Fix \(n\geq 3\) and consider \(D_n\). Let \(r\) be rotation clockwise by \(360^{\circ}/n\) and let \(s\) and \(s'\) be any two adjacent reflections of a regular \(n\)-gon. Then the following relations hold.

1. \(r^n = s^2 = (s')^2 =e\),
2. \(r^{-k} = r^{n-k}\) (special case: \(r^{-1}=r^{n-1}\)),
3. \(sr^k=r^{n-k}s\) (special case: \(sr=r^{n-1}s\)),
4. \(\underbrace{ss's\cdots}_{n\text{ factors}}=\underbrace{s'ss'\cdots}_{n\text{ factors}}\).

Exercise \(\PageIndex{2}\)

From Theorem \(\PageIndex{2}\), we know \[D_n=\langle r,s\rangle =\{\underbrace{e,r,r^2,\ldots, r^{n-1}}_{\text{rotations}},\underbrace{s,sr,sr^2,\ldots,sr^{n-1}}_{\text{reflections}}\}.\] If you were to create the group table for \(D_n\) so that the rows and columns of the table were labeled by \(e,r,r^2,\ldots, r^{n-1},s,sr,sr^2,\ldots,sr^{n-1}\) (in exactly that order), do any patterns arise? Where are the rotations? Where are the reflections?

Exercise \(\PageIndex{3}\)

What does the Cayley diagram for \(D_n\) look like if we use \(\{r,s\}\) as the generating set? What if we use \(\{s,s'\}\) as the generating set?