Definition:Symmetry Group of Square

Group Example
Let $\mathcal S = ABCD$ be a square.


 * SymmetryGroupSquare.png

The various symmetry mappings of $\mathcal S$ are:
 * The identity mapping $e$
 * The rotations $r, r^2, r^3$ of $90^\circ, 180^\circ, 270^\circ$ counterclockwise respectively about the center of $\mathcal S$.
 * The reflections $t_x$ and $t_y$ are reflections about the $x$ and $y$ axis respectively.
 * The reflection $t_{AC}$ is a reflection about the diagonal through vertices $A$ and $C$.
 * The reflection $t_{BD}$ is a reflection about the diagonal through vertices $B$ and $D$.

This group is known as the symmetry group of the square.

Also known as
The symmetry group of the square is also known as:
 * the dihedral group of order $8$ and denoted $D_4$
 * the octic group.

Some sources denote $D_4$ as ${D_4}^*$.

Also see

 * Symmetry Group of Square is Group


 * Definition:Dihedral Group