Group action

Orbit-stabilizer theorem

Given an action of a finite group on a set , for a given point the cardinality of the orbit times the order of the Stabilizer group equals the order of , i.e. group

Proof

The group acts on the set . Let . For any follows

Therefore each coset of the Stabilizer group corresponds to a different point in the orbit of , whence , and by Lagrange’s theorem, .

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