Definition:Symmetry Group of Square

Group Example

Let $\SS = ABCD$ be a square.

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 group is known as the symmetry group of the square, and can be denoted $D_4$.

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 $8$

the octic group.

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

Also see

• Symmetry Group of Square is Group

• Definition:Dihedral Group

• 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 $\text{(iv)}$

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• 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 $4$