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:


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

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:
 * $\O \in \mathcal X$

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

From Idempotent Magma Element forms Singleton Submagma:
 * $\set e \in \mathcal X$

Let us refer to the Cayley table:

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

Thus we have:

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

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

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 $\mathcal S$.