Group Action

Let be a Group and a set.
A group action is a mapping
such that for all we have:

(We use convention that means )

An action is equivalent to a Homomorphism
with
Orbit
Transitive
Stabilizer
Stabilizer of Group Action
Orbit-Stabilizer Theorem
Conjugacy Class
Centralizer
Center
Normalizer
Burnside’s Lemma

Category Theory Definition

If is a Group, a group action is a Functor to Category of Sets

It consists of a set (the singular element of is mapped to )
and mappings where