The Group of Symmetries of the Pentagon

The Group of Symmetries of the Pentagon

We have already look at The Group of Symmetries of the Equilateral Triangle and The Group of Symmetries of the Square and saw that both sets of functions describing all of their symmetries formed a group under the operation $\circ$ of function composition.

We will now see that as expected, the group of symmetries of the pentagon form a group under the operation $\circ$. Label the vertices of a pentagon with the numbers $1$, $2$, $3$, $4$, and $5$. The following table describes all ten symmetries that can be applied to the pentagon above: The top five symmetries correspond to counterclockwise rotations of $0^{\circ}$, $72^{\circ}$, $144^{\circ}$, $216^{\circ}$, and $288^{\circ}$ respectively. The bottom five symmetries corresponding to axial flips along the five axes of symmetry drawn in earn of the bottom five pentagons. We can respectively denote each of these symmetries by the functions:

(1)
\begin{align} \quad \rho_0, \rho_1, \rho_2, \rho_3, \rho_4, \mu_1, \mu_2, \mu_3, \mu_4, \mu_5 : \{ 1, 2, 3, 4, 5 \} \to \{ 1, 2, 3, 4, 5 \} \end{align}

We define each of these functions formally using Cayley's two-line notation as:

(2)
\begin{align} \quad \rho_0 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 1 & 2 & 3 & 4 & 5 \end{pmatrix} , \quad \rho_1 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 5 & 1 & 2 & 3 & 4 \end{pmatrix} , \quad \rho_2 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 4 & 5 & 1 & 2 & 3 \end{pmatrix} , \\ \rho_3 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 3 & 4 & 5 & 1 & 2 \end{pmatrix} , \quad \rho_4 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 3 & 4 & 5 & 1 \end{pmatrix} \end{align}
(3)
\begin{align} \quad \mu_1 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 1 & 5 & 4 & 3 & 2 \end{pmatrix} , \quad \mu_2 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 3 & 2 & 1 & 5 & 4 \end{pmatrix} , \quad \mu_3 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 5 & 4 & 3 & 2 & 1 \end{pmatrix} , \\ \mu_4 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 1 & 5 & 4 & 3 \end{pmatrix} , \quad \mu_5 = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 4 & 3 & 2 & 1 & 5 \end{pmatrix} \end{align}

Let $G = \{ \rho_0, \rho_1, \rho_2, \rho_3, \rho_4, \mu_1, \mu_2, \mu_3, \mu_4, \mu_5 \}$ and let $\circ : G \to G$ be defined for all $f_1, f_2 \in G$ by $(f_1 \circ f_2)(x) = f_1(f_2(x))$. Then it can be shown similarly that $(G, \circ)$ is a group.