Symmetric Groups

A group is a set of elements with a binary operation that satisfies certain mathematical axioms, allowing us to interpret operations between elements in a more general way.

Definition

Given any non-empty set $Ω$ , we define $S_{Ω}$ to be the set of all bijections from $Ω$ to itself, i.e. all permutations of $Ω$ . The set $S_{Ω}$ is a group under function composition, $\circ$ since if $σ : Ω \to Ω$ and $τ : Ω \to Ω$ are both bijections, then $σ \circ τ$ is also a bijection from $Ω$ to $Ω$ .

Axioms

Thus, all the group axioms hold for $(S_{Ω}, \circ)$ .

In the special cases where $Ω = {1, 2, \dots, n} : n \in Z$ , we refer to $S_{Ω}$ as $S_{n}$ .

\int f (x) d x = \frac{1}{2} x^{2} + C

[1324]

For any set $Ω$ , the symmetric group $S_{Ω}$ is a superset of all other maps (functions) from $Ω$ to $Ω$ , since the symmetric group captures all possible permutations and each function is a map from one such permutation to another.

Representation

We represent symmetric groups as disjoint cycles, each cycle starting with the smallest element.

Example of the $S_{3}$ group:

Value	Maps
$1$	${1, 2, 3} \mapsto {1, 2, 3}$
$(12)$	${1, 2, 3} \mapsto {2, 1, 3}$
$(13)$	${1, 2, 3} \mapsto {3, 2, 1}$
$(23)$	${1, 2, 3} \mapsto {1, 3, 2}$
$(123)$	${1, 2, 3} \mapsto {2, 3, 1}$
$(132)$	${1, 2, 3} \mapsto {2, 3, 1}$

${S}_3$ Group

key	value
$1$	$one_{1}$
2	two

Basic table

Composition of Permutations

Operations are done right to left.
$τ \circ ϕ$ is the permutation obtained by first doing $ϕ$ then doing $τ$ .

Conventionally, we avoid writing elements that are fixed by the permutations. We also order elements each cycle so the smallest element in each cycle starts.