symmetric group

Symmetric group: The symmetric group $S_n$ is the group of all permutations of $n$ elements.

For $S_n=S_{\{1,2,\dots n\}}=\{f\mid f:\{1,2,\dots n\}\to \{1,2,\dots n\},f\text{ bijective}\}$
$(S_n,\circ )$ is the symmetric group of degree $n$.
There are $|S_n|=n!$ possible permutations.

Symmetric group $S_3=\{{\text{id}}_{\{1,2,3\}},(12)(3),(13)(2),(23)(1),(123),(132)\}$

Writing out the mappings explicitly:

$$\begin{array}{rl}{\text{id}}_{\{1,2,3\}}:& 1\mapsto 1,2\mapsto 2,3\mapsto 3\\ (12)(3):& 1\mapsto 2,2\mapsto 1,3\mapsto 3\\ (13)(2):& 1\mapsto 3,2\mapsto 2,3\mapsto 1\\ (23)(1):& 1\mapsto 1,2\mapsto 3,3\mapsto 2\\ (123):& 1\mapsto 2,2\mapsto 3,3\mapsto 1\\ (132):& 1\mapsto 3,2\mapsto 1,3\mapsto 2\end{array}$$

Group Properties (using rotations as example)

Consider rotations of a square by multiples of 90°. This forms a subgroup of $S_4$ and illustrates the key group properties:

$$\begin{array}{rl}\text{Closure}:& \text{ Two rotations composed give another rotation}\\ \text{Associativity}:& \text{ Order of composing rotations doesn’t matter}\\ \text{Identity}:& \text{ Rotation by 0° leaves the square unchanged}\\ \text{Inverse}:& \text{ Each rotation has an opposite (e.g. 90° ↔ 270°)}\end{array}$$

→ $S_n$ are symmetry groups.

Every permutation in $S_n$ can be written as a product of transpositions (swaps of two elements).

Symmetry Group of the Square ( $D_4$, also $S_{\square }$)

The dihedral group $D_4$ consists of all symmetries of a square: 4 rotations and 4 reflections, giving $|D_4|=8$ elements:

$$\begin{array}{rl}\text{Rotations}:& 0°,90°,180°,270°\\ \text{Reflections}:& \text{horizontal, vertical, diagonal, anti-diagonal}\end{array}$$

Relationship between $D_4$ and $S_4$

While $S_4$ contains all possible permutations of 4 elements ($|S_4|=4!=24$), $D_4$ only contains those permutations that preserve the square’s structure. For example:
The permutation $(12)(34)$ is in $D_4$ as it represents a 180° rotation
The permutation $(123)$ is in $S_4$ but not in $D_4$ as it would “break” the square’s structure
Thus, $D_4$ is a subgroup of $S_4$