Inverse Element/Examples/Symmetry Group of Square

Examples of Inverse Elements

Consider the symmetry group of the square:

Let $\SS = ABCD$ be a square.

The various symmetries 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 $D_4$.

Each element of this group is invertible, for example:

$r^{-1} = r^3$

${t_x}^{-1} = t_x$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses