The Dihedral Groups Dn

# The Dihedral Groups Dn

Consider a regular polygon with $n$ sides. For example, a regular polygon with $3$ sides is the equilateral triangle. A regular polygon with $4$ sides is the square. A regular polygon with, say $7$ sides, is the septagon. Now consider the symmetries of this regular $n$-sided polygon. There are two types of symmetries that can arise.

The first type of symmetry that can arise are rotational symmetries by fixing the center of the polygon and rotating it counterclockwise (or clockwise) some angle. For example, the triangle, $n = 3$, can be rotated $0^{\circ}$, $120^{\circ}$, or $240^{\circ}$ around its central point and hence has $3$ rotational symmetries: The square, $n = 4$, can be rotated $0^{\circ}$, $90^{\circ}$, $180^{\circ}$, or $270^{\circ}$ around its central point and hence has $4$ rotational symmetries: In general, it's not hard to see that a regular polygon with $n$ sides will have $n$ rotational symmetries corresponding to rotations of $\left ( \frac{0}{n} \cdot 360 \right )^{\circ}$, $\left ( \frac{1}{n} \cdot 360 \right )^{\circ}$, $\left ( \frac{2}{n} \cdot 360 \right )^{\circ}$, …, $\left ( \frac{n-1}{n} \cdot 360 \right )^{\circ}$.

The second type of symmetry that can arise are axial flip symmetries. If $n$ is odd, then we can draw $n$ distinct axes that pass through each vertex of the polygon and exit the polygon by perpendicularly bisecting the opposite side to the entrance vertex and such that flipping the polygon among these axes results in a symmetry. If $n$ is even, then we can also draw $\frac{n}{2}$ distinct axes that pass through each vertex and exit through an opposite vertex and such that flipping the polygon among these axes results in a symmetry. We can also draw $\frac{n}{2}$ distinct axes that perpendicularly bisect the edge it enters and perpendicularly bisectors the opposite edge it exits from and such that flipping the polygon among these axes also results in a symmetry. In total there will be $n$ such axial flip symmetries if $n$ is even, and so in general, a polygon will have $n$ many axial symmetries.

So for each regular $n$-sided polygon there will be $n$ rotational symmetries and $n$ axial symmetries. In other words, there will be $2n$ many symmetries total. Interestingly enough, the collection of these symmetries alongside the operation of composition (i.e., composing these symmetries with one another) will always form a group. These groups are given a special name.

 Definition: The Dihedral Group for the regular $n$-sided polygon denoted $(D_n, \circ)$ is the set of symmetries $D_n$ alongside with the binary operation of composition $\circ$.

In the notation above, the subscript "$n$" on "$D_n$" denotes the number of sides on the regular polygon that we're dealing with. Some people prefer to use the subscript "$2n$" on "$D_{2n}$" to denote the number of symmetries in the group. For example, $D_6$ may refer to the dihedral group corresponding to the hexagon (using our notation) or the dihedral group corresponding to the equilateral triangle (using the later notation). We will always use the notation above and be explicit with which group we are talking about.

On the following list of pages, we will examine the dihedral groups corresponding to the equilateral triangle, square, and pentagon: