A left- group action of a Group $(G, +)$ over some set $X$ is the function

α : G \times X \to X

which satisfies the properties:

Identity: $α (e_{G}, x) \equiv e_{G} \cdot x = x$
Compatibility: $g \cdot (h \cdot x) \equiv α (g, α (h, x)) = α (g h, x) \equiv (g h) \cdot x$

Right group action

A right- group action is the function

α_{r} : X \times G \to X

which satisfies the same properties.

Properties

A group action induces an Orbit over a point $x \in X$ , which is the set

O r b_{x} = {g \cdot x \in X : \forall g \in G} \subseteq X

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

Also, every point $x \in X$ has a Stabilizer which keep it intact:

St a b_{x} = {g \in G ∣ g \cdot x = x} \subseteq G

For any point $a, b \in X$ which are connected via the group action, as $b = g \cdot a$ , then we have that

a = e \cdot a = (g^{- 1} g) \cdot x = g^{- 1} \cdot (g \cdot x) = g^{- 1} \cdot b

Thus, we have that

b = g \cdot a ⟺ a = g^{- 1} \cdot b

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 $X$ is the identity of $G$ .