# Definition:Symmetry Group of Square

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## Group Example

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

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**.

### Cayley Table

The Cayley table of the symmetry group of the square can be written:

- $\begin{array}{c|cccccc} & e & r & r^2 & r^3 & t_x & t_y & t_{AC} & t_{BD} \\ \hline e & e & r & r^2 & r^3 & t_x & t_y & t_{AC} & t_{BD} \\ r & r & r^2 & r^3 & e & t_{AC} & t_{BD} & t_y & t_x \\ r^2 & r^2 & r^3 & e & r & t_y & t_x & t_{BD} & t_{AC} \\ r^3 & r^3 & e & r & r^2 & t_{BD} & t_{AC} & t_x & t_y \\ t_x & t_x & t_{BD} & t_y & t_{AC} & e & r^2 & r^3 & r \\ t_y & t_y & t_{AC} & t_x & t_{BD} & r^2 & e & r & r^3 \\ t_{AC} & t_{AC} & t_x & t_{BD} & t_y & r & r^3 & e & r^2 \\ t_{BD} & t_{BD} & t_y & t_{AC} & t_x & r^3 & r & r^2 & e\\ \end{array}$

## 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

- Results about
**Symmetry Group of Square**can be found here.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Example $7.3$ - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.3$: Exercise $1$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Examples of groups $\text{(iv)}$

- 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups - 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups: Exercise $4$