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 $\SS$.
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 $\SS$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Example $8.5$