Group action

$Left group action$

ϕ

is a left group action of

(G, \cdot)

to

X

:

(1) ϕ (g, x) = gx (2) ϕ (1_{G}, x) = x (3) ϕ (g, ϕ (g_{2}, x)) = ϕ (g_{1} \cdot g_{2}, x)

ϕ

is right group action of

(G, \cdot)

to

X

:

(1) ϕ (g, x) = gx (2) ϕ (1_{G}, x) = x (3^{'}) ϕ (g_{1}, ϕ (g_{2}, x)) = ϕ (g_{2} \cdot g_{1}, x)

G

is a transformation group of

X

:

Action

ϕ

of

G

to

X

exists

f : X \to X x \mapsto gx

is bijective

∣ G ∣ = n \Rightarrow

Group injective homomorphism

ψ : G \to S_{n}

exists

x

is a orbit of

G

G x := {gx \in X ∣ g \in G}

Action

ϕ

is transitive:

\exists x \in X, G x = X

x

is a stabilizer group

G_{x} := {g \in G ∣ gx = x}

gx = y \Rightarrow G x = G y, G_{y} = g G_{x} g^{- 1}

x = y

x \sim_{R} y \Leftrightarrow d e f G x = G y

[x]_{R} = G x

y

is conjugate of

x

: \exists g \in G, y = gx g^{- 1}

Conjugacy class of

x

:

C (x) := {y \in G ∣\exists g \in G, y = gx g^{- 1}}

Natural bijective

π : G x \to G / G_{x}, gx \mapsto g G_{x}

exists

∣ G x ∣ = (G : G_{x}) \land ∣ G x ∣

is finite

\Rightarrow ∣ G x ∣ = ∣ G ∣/∣ G_{x} ∣

(1) ∣ C (x) ∣ = ∣ G ∣/∣ Z_{G} (x) ∣ (2) x \in Z (x) \Leftrightarrow C (x) = {x} (3) ∣ G ∣ = \sum ∣ C (x) ∣