# Closed Subsets of Symmetry Group of Square

## Theorem

Recall the symmetry group of the square:

### Symmetry Group of Square

Let $\SS = ABCD$ be a square. The various symmetry mappings of $\SS$ are:

the identity mapping $e$
the rotations $r, r^2, r^3$ of $90^\circ, 180^\circ, 270^\circ$ around the center of $\SS$ anticlockwise respectively
the reflections $t_x$ and $t_y$ are reflections in the $x$ and $y$ axis respectively
the reflection $t_{AC}$ in the diagonal through vertices $A$ and $C$
the reflection $t_{BD}$ in the diagonal through vertices $B$ and $D$.

This group is known as the symmetry group of the square, and can be denoted $D_4$.

The subsets of $\SS$ which are closed under composition of mappings are:

$\O$
$\set e$
$\set {e, r^2}$
$\set {e, t_x}$
$\set {e, t_y}$
$\set {e, t_{AC} }$
$\set {e, t_{BD} }$
$\set {e, r, r^2, r^3}$
$\set {e, r^2, t_x, t_y}$
$\set {e, r^2, t_{AC}, t_{BD} }$
$\SS$

## Proof

Recall that a submagma of an algebraic structure $\SS$ is a subsets of $\SS$ which is closed.

Let $\XX$ be the set of all submagmas of $\SS$.

$\O \in \XX$
$\SS \in \XX$
$\set e \in \XX$

Let us refer to the Cayley table:

### Cayley Table of Symmetry Group of Square

The Cayley table of the symmetry group of the square can be written:

$\begin{array}{c|cccccc}  & e & r & r^2 & r^3 & t_x & t_y & t_{AC} & t_{BD} \\  \hline e & e & r & r^2 & r^3 & t_x & t_y & t_{AC} & t_{BD} \\ r & r & r^2 & r^3 & e & t_{AC} & t_{BD} & t_y & t_x \\ r^2 & r^2 & r^3 & e & r & t_y & t_x & t_{BD} & t_{AC} \\ r^3 & r^3 & e & r & r^2 & t_{BD} & t_{AC} & t_x & t_y \\ t_x & t_x & t_{BD} & t_y & t_{AC} & e & r^2 & r^3 & r \\ t_y & t_y & t_{AC} & t_x & t_{BD} & r^2 & e & r & r^3 \\ t_{AC} & t_{AC} & t_x & t_{BD} & t_y & r & r^3 & e & r^2 \\ t_{BD} & t_{BD} & t_y & t_{AC} & t_x & r^3 & r & r^2 & e\\ \end{array}$

Taking each of the elements of $\SS$ in order:

 $\ds t_x \circ t^x$ $=$ $\ds e$ $\ds t_y \circ t^y$ $=$ $\ds e$ $\ds t_{AC} \circ t^{AC}$ $=$ $\ds e$ $\ds t_{BD} \circ t^{BD}$ $=$ $\ds e$ $\ds r^2 \circ r^2$ $=$ $\ds e$ $\ds r \circ r$ $=$ $\ds r^2$ $\ds r \circ r^2$ $=$ $\ds r^3$ $\ds r \circ r^3$ $=$ $\ds e$ $\ds r^3 \circ r^3$ $=$ $\ds r^2$ $\ds r^3 \circ r^2$ $=$ $\ds r$ $\ds r^3 \circ r$ $=$ $\ds e$

Thus we have:

 $\ds \set {e, r^2}$ $\in$ $\ds \XX$ $\ds \set {e, t_x}$ $\in$ $\ds \XX$ $\ds \set {e, t_y}$ $\in$ $\ds \XX$ $\ds \set {e, t_{AC} }$ $\in$ $\ds \XX$ $\ds \set {e, t_{BD} }$ $\in$ $\ds \XX$ $\ds \set {e, r, r^2, r^3}$ $\in$ $\ds \XX$

Next note by inspection that:

$\set {e, r^2, t_x, t_y} \in \XX$

and:

$\set {e, r^2, t_{AC}, t_{BD} } \in \XX$
 $\ds \set {e, r^2, t_x, t_y}$ $\in$ $\ds \XX$ $\ds \set {e, r^2, t_{AC}, t_{BD} }$ $\in$ $\ds \XX$

Finally note by inspection that:

any closed subset of $\SS$ which contains both $r$ and any of the reflections contains all the elements of $\SS$
any closed subset of $\SS$ which contains both $r^3$ and any of the reflections contains all the elements of $\SS$.

Thus there are no more proper subsets of $\SS$ which are submagmas of $\SS$.

$\blacksquare$