Symmetry Group of Square is Group

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Theorem

The symmetry group of the square is a non-abelian group.


Definition

Recall the definition of the symmetry group of the square:


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.


Proof

Let us refer to this group as $D_4$.


Taking the group axioms in turn:


Group Axiom $\text G 0$: Closure

From the Cayley table it is seen directly that $D_4$ is closed.

$\Box$


Group Axiom $\text G 1$: Associativity

Composition of Mappings is Associative.

$\Box$


Group Axiom $\text G 2$: Existence of Identity Element

The identity is $e$ as defined.

$\Box$


Group Axiom $\text G 3$: Existence of Inverse Element

Each element can be seen to have an inverse:

$r^{-1} = r^3$ and so $\paren {r^3}^{-1} = r$
$r^2$, $t_{AC}$, $t_{BD}$, $t_x$ and $t_y$ are all self-inverse.

$\Box$


Thus $D_4$ is seen to be a group.

Note that from the Cayley table it can be observed directly that:

$r \circ t_x = t_{BD}$
$t_x \circ r = t_{AC}$

thus illustrating by counterexample that $D_4$ is not abelian.

$\blacksquare$


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