Orbit counting lemma

Given an action of the group $𝐺$ on the set $Ω$ , let $F i x Ω (𝑔)$ denote the set of all $𝜔 \in Ω$ left invariant by $𝑔 \in 𝐺$ . Then the number of orbits of $𝐺$ in $Ω$ is group

| Ω / 𝐺 | = 1 | 𝐺 | \sum 𝑔 \in 𝐺 | F i x Ω (𝑔) | = 1 | 𝐺 | 𝑟 \sum 𝑖 = 1 ∣ 𝑔 𝑖 𝐺 ∣ | F i x Ω (𝑔) |

where $𝑔 𝑖 𝐺$ enumerate conjugacy classes.

Proof

proof

develop | en | SemBr

Orbit counting lemma

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