Closed Subsets of Symmetry Group of Square

Theorem
Recall the symmetry group of the square:

Symmetry Group of Square
The subsets of $\mathcal S$ which are closed under composition of mappings are:


 * $\varnothing$
 * $\left\{ {e}\right\}$
 * $\left\{ {e, r^2}\right\}$
 * $\left\{ {e, t_x}\right\}$
 * $\left\{ {e, t_y}\right\}$
 * $\left\{ {e, t_{AC} }\right\}$
 * $\left\{ {e, t_{BD} }\right\}$
 * $\left\{ {e, r, r^2, r^3}\right\}$
 * $\left\{ {e, r^2, t_x, t_y}\right\}$
 * $\left\{ {e, r^2, t_{AC}, t_{BD} }\right\}$
 * $\mathcal S$

Proof
Recall that a submagma of an algebraic structure $\mathcal S$ is a subsets of $\mathcal S$ which is closed.

Let $\mathcal X$ be the set of all submagmas of $\mathcal S$.

From Empty Set is Submagma of Magma:
 * $\varnothing \in \mathcal X$

From Magma is Submagma of Itself:
 * $\mathcal S \in \mathcal X$

From Idempotent Magma Element forms Singleton Submagma:
 * $\left\{ {e}\right\} \in \mathcal X$

Let us refer to the Cayley table:

Cayley Table of Symmetry Group of Square
Taking each of the elements of $\mathcal S$ in order:


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

Thus we have:
 * $\left\{ {e, r^2}\right\} \in \mathcal X$
 * $\left\{ {e, t_x}\right\} \in \mathcal X$
 * $\left\{ {e, t_y}\right\} \in \mathcal X$
 * $\left\{ {e, t_{AC} }\right\} \in \mathcal X$
 * $\left\{ {e, t_{BD} }\right\} \in \mathcal X$
 * $\left\{ {e, r, r^2, r^3}\right\} \in \mathcal X$

Next note by inspection that:
 * $\left\{ {e, r^2, t_x, t_y}\right\} \in \mathcal X$

and:
 * $\left\{ {e, r^2, t_{AC}, t_{BD} }\right\} \in \mathcal X$

Finally note by inspection that:
 * any closed subset of $\mathcal S$ which contains both $r$ and any of the reflections contains all the elements of $\mathcal S$
 * any closed subset of $\mathcal S$ which contains both $r^3$ and any of the reflections contains all the elements of $\mathcal S$.

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