Orbit Stabilizer Theorem: Statement, Proof

The orbit-stabilizer theorem of groups says that the size of a finite group G is the multiplication of the size of the orbit of an element a (in A on which G acts) with that of the stabilizer of a. In this article, we will learn about what are orbits and stabilizers. We will also explain the orbit-stabilizer theorem in detail with proof.

What are Orbits and Stabilizers

Let G be a group and let A be a finite set on which G acts. That is, we have a map

G×A → A

defined by (g, a) $\mapsto$ g⋅a for all g∈G and a∈A.

Definition of Orbits: The orbit of an element a∈A is denoted by G⋅a and defined by

G⋅a = {g⋅a: g ∈ G}.

Definition of Stabilizers: The stabilizer of an element a ∈ A, denoted by G_a, is defined as follows

G_a = {g ∈ G: g⋅a=a}.

Orbit Stabilizer Theorem

Statement: If G is a finite group acting on a finite set A, then |G| = |G⋅a| × |G_a| for a∈A. That is,

$|G \cdot a|=\dfrac{|G|}{G_a}$.

Orbit Stabilizer Theorem Proof

We define a mapping φ: G → G⋅a by

φ(g) = g⋅a ∀ g∈G.

Now for g, h ∈ G, we have

φ(g) = φ(h) ⇔ g⋅a = h⋅a ⇔ g^-1h⋅a=a ⇔ g^-1h∈G_a ⇔ h∈gG_a

This shows that both g and h lie in the same coset for the subgroup G_a and hence φ induces a bijective mapping

$G/G_a \xrightarrow{\sim} G\cdot a$.

As G is a finite group, applying Lagrange’s theorem we complete the proof of the orbit-stabilizer theorem which is

$\dfrac{|G|}{|G_a|} = |G\cdot a|$.

Related Topics:

Group Theory: Definition, Examples, Properties

Abelian Group: Definition, Properties, Examples

Cyclic Group: Definition, Orders, Properties, Examples

Normal Subgroup

Quotient Group

Kernel of a Group Homomorphism

First Isomorphism Theorem

Center of a Group

FAQs on Orbit Stabilizer Theorem