Definition:Symmetry Group of Square

Group Example
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.

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