# Orbit-Stabilizer Theorem

Let be a group which acts on a finite set . Then

## Proof

Define by

Well-defined:

Note that is a subgroup of . If , then . Thus , which implies , thus is well-defined.

Surjective:

is clearly surjective.

Injective:

If , then . Thus , so . Thus .

By Lagrange’s Theorem,

Field Medallist Prof. Gowers has also written a nice post on the Orbit -Stabilizer Theorem and various proofs.

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