Group Actions and Sylow Theorems - Main Theorem

The orbit-stabilizer theorem is the fundamental counting principle for group actions, relating orbits and stabilizers through a beautiful equation.

TheoremOrbit-Stabilizer Theorem

Let $G$ be a finite group acting on a set $X$ , and let $x \in X$ . Then: $|G| = |\text{Orb}(x)| \cdot |\text{Stab}(x)|$

Equivalently, the size of the orbit of $x$ equals the index of its stabilizer: $|\text{Orb}(x)| = [G : \text{Stab}(x)] = \frac{|G|}{|\text{Stab}(x)|}$

This theorem generalizes Lagrange's theorem. The orbit measures how many distinct positions $x$ can be moved to, while the stabilizer measures the symmetry fixing $x$ . Their product always equals $|G|$ .

ProofProof Sketch

Define a map $\phi: G/\text{Stab}(x) \to \text{Orb}(x)$ by $\phi(g \cdot \text{Stab}(x)) = g \cdot x$ . One can verify:

Well-defined: If $g_1 \cdot \text{Stab}(x) = g_2 \cdot \text{Stab}(x)$ , then $g_1^{-1}g_2 \in \text{Stab}(x)$ , so $g_1 \cdot x = g_2 \cdot x$
Injective: If $g_1 \cdot x = g_2 \cdot x$ , then $g_1^{-1}g_2 \cdot x = x$ , so $g_1^{-1}g_2 \in \text{Stab}(x)$
Surjective: Every element of $\text{Orb}(x)$ has form $g \cdot x$ for some $g \in G$

Thus $\phi$ is a bijection, giving $|G/\text{Stab}(x)| = |\text{Orb}(x)|$ .

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ExampleRotation Groups

Consider the rotation group $SO(3)$ acting on the unit sphere $S^2$ . For any point $p \in S^2$ :

$\text{Orb}(p) = S^2$ (the full sphere—we can rotate $p$ to any other point)
$\text{Stab}(p) \cong SO(2)$ (rotations fixing $p$ are rotations about the axis through $p$ )

The orbit-stabilizer theorem gives: $|SO(3)| = |S^2| \cdot |SO(2)|$ (informally, relating the "sizes" of these continuous groups).

TheoremCauchy's Theorem

If $G$ is a finite group and $p$ is a prime dividing $|G|$ , then $G$ contains an element of order $p$ .

This follows from group actions: consider $G$ acting on $G^p$ by cyclic permutation. Counting fixed points modulo $p$ yields the result.

Remark

The orbit-stabilizer theorem underpins most counting arguments in group theory. It appears in the class equation, Burnside's lemma, and the proof of the Sylow theorems. Understanding this single theorem unlocks a vast array of applications.

The interplay between orbits (global structure) and stabilizers (local symmetry) reveals deep connections between group theory and geometry.