Definitions. Let X be a set and G be a group. A left group action of G on X is a map G x X → X given by (g, x) → g·x such that the following axioms hold:
a) Identity: ∀x ∈ X, e·x = x
b) Compatibility: ∀x ∈ X, ∀g, h ∈ G, g·(h·x) = (gh)·x
The group G is said to act on the set X. A set X equipped with an action by G is called a G-set.
Let X be a set and G be a group. A right group action of G on X is a map X x G→ X given by (x, g) → x·g such that the following axioms hold:
a) Identity: ∀x ∈ X, x·e = x
b) Compatibility: ∀x ∈ X, ∀g, h ∈ G, (x·g)·h = x·(gh)
Examples.
a) ∀x ∈ ℝ^{n}, I_{n}x = x
b) ∀x ∈ ℝ^{n}, ∀A, B ∈ G = GL_{n}(ℝ), A(Bx) = (AB)x.
This is called the matrix action.
a) ∀x ∈ X, 1·x = Id(x) = x b) ∀x ∈ X, ∀σ, τ ∈ S_{n}: σ·(τ·x) = σ·(τ(x)) = σ(τ(x)) = (σ∘τ)(x) = (στ)·x
This is called the permutation action.
Cayley’s Theorem states that every group is isomorphic to a permutation group, i.e., a subgroup of a symmetric group, therefore up to isomorphism every group action is a permutation action.
A group G acts on itself by left multiplication, that is, g·x = gx. This is called the left regular action. Viewing the regular action as a permutation action, we have g·x = λ_{g}(x), where λ_{g}: G → G is the left multiplication map defined as x → gx
A group G acts on itself by conjugation, that is, g, x ∈ G, and define g·x = gxg^{-1}.
a) e·x = exe^{-1} = x.
b) ∀ g, h ∈ G. g·(h·x)= g·(hxh^{-1}) = g(hxh^{-1})g^{-1}= [shoe-sock theorem] (gh)x(gh)^{-1} = (gh)·x
This is called the conjugation action.
Notice that X does not need to be a subgroup, so H does not need to be a normal subgroup.
Theorem. Let X be a G-set. Then define a relation ~ on X, by the rule x ~ y if ∃g ∈ G: g·x = y. Then ~ is an equivalence relation on X, called G-equivalence. The equivalence classes of this relation are called orbits, where O_{x} is the orbit containing x ∈ X.
Proof.
Recall, a set X equipped with an action by G is called a G-set. Claim: ~ is an equivalence relation on X
[G is a group ⇒ ∃g^{-1}∈G: g^{-1}g = gg^{-1} = e] g^{-1}·y = g^{-1}·(g·x) = (g^{-1}g)·x = e·x = x ⇒ y ~ x.
Example. Let G the permutation group G = ⟨(123), (45)⟩ = ⟨(123)⟩⟨(45)⟩ = {1, (123), (132), (45), (123)(45), (132)(45)} ≤ S_{5}, and let X = {1, 2, 3, 4, 5}. Then, G acts a permutation on X (σ·x = σ(x) for x ∈S_{n})
O_{1} = O_{2} = O_{3} = {1, 2, 3} e.g., let σ = (123), x= 1, 1 ~ 2 because σ∈G: σ·1=σ(1)= 2; 2 ~ 3 because σ∈G: σ·2=σ(2)=3. τ = (132), τ·1 = τ(1) = 3, so 1 ~ 3. Therefore, [1]_{~} = O_{1} = O_{2} = O_{3} = {1, 2, 3}
O_{4} = O_{5} = {4, 5}.
The Fundamental Counting Principle. Let G be a group acting on X and x an element of X. Then, the cardinality of an orbit is equal to the index of its stabilizer, that is, |O_{x}| = [G:G_{x}]
Recall: The stabilizer of x is the set G_{x} = { g ∈ G ∣ g ⋅ x = x} , the set of elements of G which leave x unchanged under the action.
Proof.
Reclaim: O_{x} = {y ∈ X | y = g·x}, G_{x} = {g ∈ X | g·x = x}
It suffices to show a bijection between O_{x} and G/G_{x}. Notice that if y ∈ O_{x} ⇒ ∃g ∈ X | y = g·x ⇒ g·x denotes an arbitrary element of O_{x}
Let’s define the map f: O_{x} → G/G_{x} by the rule, f(g·x) = gG_{x}
Remark. If G is a finite group, then |O_{x}| = [G:G_{x}] = |G|/|G_{x}| and therefore, |G| = |O_{x}||G_{x}|, and the cardinality of the orbit divides the cardinality of the group.