Symmetric Group

algebra

Definition

Symmetric Group

A symmetric group is a group of all permutations of a finite set. The symmetric group on a set of size $n$ is denoted by $S_n$ and it represents the collection of all possible ways to arrange $n$ distinct elements.

The symmetric group $S_n$ is the group of all bijective functions (permutations) from a set $\{1,2,\dots ,n\}$ to itself, under the operation of function composition.

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Examples

$S_1$

The symmetric group on a single element has only one element, the identity permutation. Thus $S_1\cong \{{\text{id}}_{\{a\}}\}$ (where $e$ is the neutral element).

$S_2$

The symmetric group on two elements has two permutations:

• Identity: ${\text{id}}_{\{1,2\}}=(1)(2)$
• Transposition: $(12)$, which swaps the two elements. Hence, $S_2\cong {\mathbb{Z}}_2$.

$S_3$

The symmetric group on three elements has $3!=6$ elements:

• Identity: ${\text{id}}_{\{1,2,3\}}=(1)(2)(3)$
• Transposition: $(12),(13),(23)$
• Cycles: $(123),(132)$

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

$S_4$

The symmetric group on four elements has $4!=24$ elements, consisting of all permutations of $1,2,3,4$.

Square

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Composition

Compositions can easily be done in cycle notation:

$$\pi \circ \sigma :(1)(3)(24)\circ (1234)=(14)(23)$$ $$\begin{array}{rlr}1.& \sigma :1\mapsto 2\pi :2\mapsto 4& ⟹\pi \circ \sigma :1\mapsto 4\\ 2.& \sigma :2\mapsto 3\pi :3\mapsto 3& ⟹\pi \circ \sigma :2\mapsto 3\\ 3.& \sigma :3\mapsto 4\pi :4\mapsto 2& ⟹\pi \circ \sigma :3\mapsto 2\\ 4.& \sigma :4\mapsto 1\pi :1\mapsto 1& ⟹\pi \circ \sigma :4\mapsto 1\end{array}$$