# 5.2: Dihedral Groups

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Another special type of permutation group is the dihedral group. Recall the symmetry group of an equilateral triangle in Chapter 3. Such groups consist of the rigid motions of a regular $$n$$-sided polygon or $$n$$-gon. For $$n = 3, 4, \ldots\text{,}$$ we define the nth dihedral group to be the group of rigid motions of a regular $$n$$-gon. We will denote this group by $$D_n\text{.}$$ We can number the vertices of a regular $$n$$-gon by $$1, 2, \ldots, n$$ (Figure 5.19). Notice that there are exactly $$n$$ choices to replace the first vertex. If we replace the first vertex by $$k\text{,}$$ then the second vertex must be replaced either by vertex $$k+1$$ or by vertex $$k-1\text{;}$$ hence, there are $$2n$$ possible rigid motions of the $$n$$-gon. We summarize these results in the following theorem.

$$Figure \text { } 5.19.$$ A regular $$n$$-gon

The dihedral group, $$D_n\text{,}$$ is a subgroup of $$S_n$$ of order $$2n\text{.}\ ##### Theorem \(5.21$$

The group $$D_n\text{,}$$ $$n \geq 3\text{,}$$ consists of all products of the two elements $$r$$ and $$s\text{,}$$ satisfying the relations

\begin{align*} r^n & = 1\\ s^2 & = 1\\ srs & = r^{-1}\text{.} \end{align*}
Proof

The possible motions of a regular $$n$$-gon are either reflections or rotations (Figure 5.22). There are exactly $$n$$ possible rotations:

$\identity, \frac{360^{\circ} }{n}, 2 \cdot \frac{360^{\circ} }{n}, \ldots, (n-1) \cdot \frac{360^{\circ} }{n}\text{.} \nonumber$

We will denote the rotation $$360^{\circ} /n$$ by $$r\text{.}$$ The rotation $$r$$ generates all of the other rotations. That is,

$r^k = k \cdot \frac{360^{\circ} }{n}\text{.} \nonumber$
$$Figure \text { } 5.22.$$ Rotations and reflections of a regular $$n$$-gon

Label the $$n$$ reflections $$s_1, s_2, \ldots, s_n\text{,}$$ where $$s_k$$ is the reflection that leaves vertex $$k$$ fixed. There are two cases of reflections, depending on whether $$n$$ is even or odd. If there are an even number of vertices, then two vertices are left fixed by a reflection, and $$s_1 = s_{n/2 + 1}, s_2 = s_{n/2 + 2}, \ldots, s_{n/2} = s_n\text{.}$$ If there are an odd number of vertices, then only a single vertex is left fixed by a reflection and $$s_1, s_2, \ldots, s_n$$ are distinct (Figure 5.23). In either case, the order of each $$s_k$$ is two. Let $$s = s_1\text{.}$$ Then $$s^2 = 1$$ and $$r^n = 1\text{.}$$ Since any rigid motion $$t$$ of the $$n$$-gon replaces the first vertex by the vertex $$k\text{,}$$ the second vertex must be replaced by either $$k+1$$ or by $$k-1\text{.}$$ If the second vertex is replaced by $$k+1\text{,}$$ then $$t = r^k\text{.}$$ If the second vertex is replaced by $$k-1\text{,}$$ then $$t = r^k s\text{.}$$ 2  Hence, $$r$$ and $$s$$ generate $$D_n\text{.}$$ That is, $$D_n$$ consists of all finite products of $$r$$ and $$s\text{,}$$

$D_n = \{1, r, r^2, \ldots, r^{n-1}, s, rs, r^2 s, \ldots, r^{n-1} s\}\text{.} \nonumber$

We will leave the proof that $$srs = r^{-1}$$ as an exercise.

Since we are in an abstract group, we will adopt the convention that group elements are multiplied left to right.
$$Figure \text { } 5.23.$$ Types of reflections of a regular $$n$$-gon

This page titled 5.2: Dihedral Groups is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Thomas W. Judson (Abstract Algebra: Theory and Applications) via source content that was edited to the style and standards of the LibreTexts platform.