Definition
Symmetric Group
A symmetric group is a group of all permutations of a finite set. The symmetric group on a set of size
is denoted by and it represents the collection of all possible ways to arrange distinct elements. The symmetric group
is the group of all bijective functions (permutations) from a set to itself, under the operation of function composition.
Examples
The symmetric group on a single element has only one element, the identity permutation. Thus
The symmetric group on two elements has two permutations:
- Identity:
- Transposition:
, which swaps the two elements. Hence, .
The symmetric group on three elements has
- Identity:
- Transposition:
- Cycles:
The symmetric group on four elements has
Square
Composition
Compositions can easily be done in cycle notation:
