Symmetric Group
The symmetric group is the set of permutations of , with composition as its group operation.
Definition 1.14 (Symmetric Group)
is defined to be the set of permutations of a set .
We will prove that this is a group in Proposition 1.16.
Group Structure
Proposition 1.16
For any set , is a group.
Proof
Composition of permutations is a permutation, composition is associative, the identity map is the identity element, and every permutation has an inverse by the definition of a bijection. Thus is a group.
Stated in
- Definition 1.14 (Symmetric Group)§1.5 Symmetric Groups
- Proposition 1.16§1.5 Symmetric Groups