Orbit-stabilizer theorem

Given an action of a finite group $𝐺$ on a set $𝑀$ , for a given point $𝑚 \in 𝑀$ the cardinality of the orbit times the order of the Stabilizer group equals the order of $𝐺$ , i.e. #m/thm/group

| 𝐺 𝑚 | \cdot | 𝐺 𝑚 | = | 𝐺 |

Proof

The group $𝐺$ acts on the set $𝑀$ . Let $𝑚 \in 𝑀$ . For any $𝑔 1, 𝑔 2 \in 𝐺$ follows

𝑔 1 𝑚 = 𝑔 2 𝑚 ⟺ 𝑚 = 𝑔 - 1 1 𝑔 2 𝑚 ⟺ 𝑔 - 1 1 𝑔 2 \in 𝐺 𝑚 ⟺ 𝑔 2 \in 𝑔 1 𝐺 𝑚

Therefore each coset of the Stabilizer group $𝐺 𝑚$ corresponds to a different point in the orbit of $𝑚$ , whence $| 𝐺 𝑚 | = [𝐺 𝑚 : 𝐺]$ , and by Lagrange's theorem, $| 𝐺 𝑚 | \cdot | 𝐺 𝑚 | = | 𝐺 |$ .

#state/tidy | #lang/en | #SemBr