Right group action

A right- group action is the function

which satisfies the same properties.

Properties

A group action induces an Orbit over a point , which is the set

Note that Group actions partition a set into its orbits, since their connection via a group element induces an Equivalence Relation.

Also, every point has a Stabilizer which keep it intact:

For any point which are connected via the group action, as , then we have that

Thus, we have that

A group action is a Transitive Group Action if it’s a group action which allows transitivity between points in the set.

A group action is said to be a Free Group Action if the only element which stabilizes all elements in is the identity of .