Theorem - Orbit Stabilizer Formula

Formal Statement

For a group $G$ acting on a set $X$ , with $G, X$ finite, for all $x \in X$ ,

∣ orb_{G} (x) ∣ = \frac{∣ G ∣}{∣ stab _{G} ( x ) ∣} .

Proof

Say $G = {g_{1}, \dots, g_{n}}$ .

$orb (x) = {g_{1} ∙ x, \dots, g_{n} ∙ x}$ . From this, we have that there are at most $∣ G ∣$ elements, but that there very well may be repetitions.

When is $g_{i} ∙ x = g_{j} ∙ x$ ?

Precisely when $(g_{j}^{- 1} g_{i}) ∙ x = x ⟺ g_{j}^{- 1} g_{i} \in stab (x) ⟺ g_{i} \in g_{j} stab (x)$ ← left coset. Fixing $i$ , $# {j : g_{j} ∙ x = g_{i} ∙ x} = ∣ g_{j} stab (x) ∣ = ∣ stab (x) ∣ ⟹$ on list, $g_{1} ∙ x, \dots, g_{n} ∙ x$ ( $∣ G ∣$ elements), each element repeats $∣ stab (x) ∣$ times.

$⟹ ∣ {g_{1} ∙ x, \dots, g_{n} ∙ x} ∣ = \frac{∣ G ∣}{∣ stab ( x ) ∣} . □$

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Theorem - Orbit Stabilizer Formula

Formal Statement

Proof

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