Closed Subsets of Symmetry Group of Square

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Theorem

Recall the symmetry group of the square:

Symmetry Group of Square

Let $\SS = ABCD$ be a square.

SymmetryGroupSquare.png

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 center is known as the symmetry group of the square.


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 $\SS$.

From Empty Set is Submagma of Magma:

$\O \in \XX$

From Magma is Submagma of Itself:

$\SS \in \XX$

From Idempotent Magma Element forms Singleton Submagma:

$\set e \in \XX$


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 $\SS$.

$\blacksquare$


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