Orbit-Stabilizer Theorem (with proof)

Orbit-Stabilizer Theorem

Let G be a group which acts on a finite set X. Then \displaystyle |\text{Orb}(x)|=[G:\text{Stab}(x)]=\frac{|G|}{|\text{Stab}(x)|}.

Proof

Define \phi:G/\text{Stab}(x)\to\text{Orb}(x) by \displaystyle \phi(g\text{Stab}(x))=g\cdot x.

Well-defined:

Note that \text{Stab}(x) is a subgroup of G. If g\text{Stab}(x)=h\text{Stab}(x), then g^{-1}h\in\text{Stab}(x). Thus g^{-1}hx=x, which implies hx=gx, thus \phi is well-defined.

Surjective:

\phi is clearly surjective.

Injective:

If \phi(g\text{Stab}(x))=\phi(h\text{Stab}(x)), then gx=hx. Thus g^{-1}hx=x, so g^{-1}h\in\text{Stab}(x). Thus g\text{Stab}(x)=h\text{Stab}(x).

By Lagrange’s Theorem, \displaystyle \frac{|G|}{|\text{Stab}(x)|}=|G/\text{Stab}(x)|=|\text{Orb}(x)|.

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

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