Question

List the elements of the group ${\mathit{D}}_{\mathbf{5}}$(the symmetries of a regular pentagon). [Hint: The group has order 10.]

Short answer

The five rotations of $D_5$are given by,$\left[\begin{array}{cc}\cos \frac{i2\pi }{5}& \sin \frac{i2\pi }{5}\\ -\sin \frac{i2\pi }{5}& \cos \frac{i2\pi }{5}\end{array}\right]$, where $i\in \{0,1,2,3,4\}$.

And, its five reflections are given by,$\left[\begin{array}{cc}\cos \frac{i2\pi }{5}& \sin \frac{i2\pi }{5}\\ \sin \frac{i2\pi }{5}& -\cos \frac{i2\pi }{5}\end{array}\right]$, where$i\in \{0,1,2,3,4\}$.

Step-by-step solution

Definition ofD5D5D5

It is defined as symmetries of a regular polygon (here, pentagon), whose center is fixed at origin formed by rotations and reflections.

Finding elements of the group D5 

The rotational symmetries of$D_5$are given by dividing the complete angle $2\pi$into five parts.

That means, there are five rotational symmetries around the centre given by the multiples of,$\frac{2\pi }{5}rad=72°$and the reflections are across the bisector of each angle.

Let there be a regular pentagon in 2-dimensional plane with centre at the origin $(0,0)$and one of its vertex being at $(1,0)$. Then, its five rotations are given by,

localid="1659418373757" $\left[\begin{array}{cc}\cos \frac{i2\pi }{5}& \sin \frac{i2\pi }{5}\\ -\sin \frac{i2\pi }{5}& \cos \frac{i2\pi }{5}\end{array}\right]$, wherelocalid="1659418377858" $i\in \{0,1,2,3,4\}$

And, its five reflections are given by,localid="1659418392810" $\left[\begin{array}{cc}\cos \frac{i2\pi }{5}& \sin \frac{i2\pi }{5}\\ \sin \frac{i2\pi }{5}& -\cos \frac{i2\pi }{5}\end{array}\right]$, where localid="1659418397092" $i\in \{0,1,2,3,4\}$.