Dihedral groups are groups of symmetries of regular $n$-gons. We start with an example.

##### Example6.5.1

Consider a regular triangle $T\text{,}$ with vertices labeled $1\text{,}$ $2\text{,}$ and $3\text{.}$ We show $T$ below, also using dotted lines to indicate a vertical line of symmetry of $T$ and a rotation of $T\text{.}$

Note that if we reflect $T$ over the vertical dotted line (indicated in the picture by $f$), $T$ maps onto itself, with $1$ mapping to $1\text{,}$ and $2$ and $3$ mapping to each other. Similarly, if we rotate $T$ clockwise by $120^{\circ}$ (indicated in the picture by $r$), $T$ again maps onto itself, this time with $1$ mapping to $2\text{,}$ $2$ mapping to $3\text{,}$ and $3$ mapping to $1\text{.}$ Both of these maps are called symmetries of $T\text{;}$ $f$ is a reflection or flip and $r$ is a rotation.

Of course, these are not the only symmetries of $T\text{.}$ If we compose two symmetries of $T\text{,}$ we obtain a symmetry of $T\text{:}$ for instance, if we apply the map $f\circ r$ to $T$ (meaning first do $r\text{,}$ then do $f$) we obtain reflection over the line connecting $2$ to the midpoint of line segment $\overline{13}\text{.}$ Similarly, if we apply the map $f\circ (r\circ r)$ to $T$ (first do $r$ twice, then do $f$) we obtain reflection over the line connecting $3$ to the midpoint of line segment $\overline{12}\text{.}$ In fact, every symmetry of $T$ can be obtained by composing applications of $f$ and applications of $r\text{.}$

For convenience of notation, we omit the composition symbols, writing, for instance, $fr$ for $f\circ r\text{,}$ $r\circ r$ as $r^2\text{,}$ etc. It turns out there are exactly six symmetries of $T\text{,}$ namely:

1. the map $e$ from $T$ to $T$ sending every element to itself;

2. $f$ (that is, reflection over the line connecting $1$ and the midpoint of $\overline{23}$);

3. $r$ (that is, clockwise rotation by $120^{\circ}$);

4. $r^2$ (that is, clockwise rotation by $240^{\circ}$);

5. $fr$ (that is, reflection over the line connecting $2$ and the midpoint of $\overline{13}$); and

6. $fr^2$ (that is, reflection over the line connecting $3$ and the midpoint of $\overline{12}$).

Declaring that $f^0=r^0=e\text{,}$ the set

\begin{equation*} D_3=\{e, f, r, r^2, fr, fr^2\}=\{f^ir^j:i=0,1, j=0,1,2\} \end{equation*}

is the collection of all symmetries of $T\text{.}$

##### Remark6.5.2

Notice that $rf=fr^2$ and that $f^2=r^3=e\text{.}$

##### Proof

Let us look at $D_3$ another way. Note that each map in $D_3$ can be uniquely described by how it permutes the vertices $1,2,3$ of $T\text{:}$ that is, each map in $D_3$ can be uniquely identified with a unique element of $S_3\text{.}$ For instance, $f$ corresponds to the permutation $(23)$ in $S_3\text{,}$ while $fr$ corresponds to the permutation $(13)\text{.}$ In turns out that $D_3 \simeq S_3\text{,}$ via the following correspondence.

 $e$ $\mapsto$ $e$ $f$ $\mapsto$ $(23)$ $r$ $\mapsto$ $(123)$ $r^2$ $\mapsto$ $(132)$ $fr$ $\mapsto$ $(13)$ $fr^2$ $\mapsto$ $(12)$

The group $D_3$ is an example of class of groups called dihedral groups.

##### Definition6.5.4

Let $n$ be an integer greater than or equal to $3\text{.}$ We let $D_n$ be the collection of symmetries of the regular $n$-gon. It turns out that $D_n$ is a group (see below), called the dihedral group of order $2n$. (Note: Some books and mathematicians instead denote the group of symmetries of the regular $n$-gon by $D_{2n}$—so, for instance, our $D_3\text{,}$ above, would instead be called $D_6\text{.}$ Make sure you are aware of the convention your book or colleague is using.)

##### Remark6.5.6

Throughout this course, if we are discussing a group $D_n$ you should assume $n\in \Z^+\text{,}$ $n\geq 3\text{,}$ unless otherwise noted.

##### Definition6.5.7

We say that an element of $D_n$ is written in standard form if it is written in the form $f^ir^j$ where $i\in \{0,1\}$ and $j\in \{0,1,\ldots,n-1\}\text{.}$

##### Warning6.5.9

While $D_3$ is actually isomorphic to $S_3$ itself, for $n>3$ we have that $D_n$ is not isomorphic to $S_n$ but is rather isomorphic to a proper subgroup of $S_n\text{.}$ When $n>3$ you can see that $D_n$ cannot be isomorphic to $S_n$ since $|D_n|=2n \lt n! = |S_n|$ for $n>3\text{.}$

It is important to be able to do computations with specific elements of dihedral groups. We have the following theorem.

##### Example6.5.11

1. Write $fr^2f$ in $D_3$ in standard form. Do the same for $fr^2f$ in $D_4\text{.}$

2. What is the inverse of $fr^3$ in $D_5\text{?}$ Write it in standard form.

3. Explicitly describe an isomorphism from $D_4$ to a subgroup of $S_4\text{.}$

##### Example6.5.12

Classify the following groups up to isomorphism. (Hint: You may want to look at the number of group elements that have a specific finite order.)

\begin{equation*} \Z, \Z_6, \Z_2, S_6, \Z_4, \Q, 3\Z, \R, S_2, \R^*, S_3,\Q^*, \C^*, \langle \pi\rangle \text{ in } \R^*, \end{equation*} \begin{equation*} D_6, \langle (134)(25)\rangle \text{ in } S_5, \R^+, D_3, \langle r \rangle \text{ in } D_4, 17\Z \end{equation*}