Group actions

Let be a group and any set. An action or operation of on is a map such that

1. for every .
2. for every and .

A group action can be seen as just a homomorphism from to . This is because for any , the map is a bijection since it has the inverse . So the map that takes to the map is a homomorphism from to .

Let be a group, any set, and be a group action by on . The stabilizer subgroup or isotropy subgroup of in is The -orbit of is and the orbit space is Furthermore, we call the group action transitive if .

We can define an equivalence relation on by This makes the orbits of equivalence classes with respect to :Thus we see that is the disjoint union of its orbits.

Let be a group, any set and a group action by on . If is finite,

Let be a group, any set, and be a group action by on . Then the -map induces a bijection where is the stabilizer of . Furthermore, Proof:
The map given by is clearly surjective. For any we see that This means that the map given by is well-defined, surjective and injective, so a bijection.

Let be a group, any set, and be a group action by on . Then

Let be a finite group, any set and . Then Proof:
Let . We then have and meaning that The sizes of the left and right expression being the same then gives from which the statement follows directly.