$G<S_n$ is transitive
calculate $1/|G| * \sum_{g \in G} f(g)$
where $G<S_n$ and $f(g) = |\{ 1 \le i \le n | g(i) = i \}|$
I tried to use the orbit stabiliser theorem but didn't get anywhere
any hint or solution will be very appriciated
2026-04-01 14:45:07.1775054707
If $G<S_n$ is transitive, calculate $1/|G| \cdot \sum_{g \in G} f(G)$
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By Burnside's lemma, $1/|G| \cdot \sum_{g \in G} f(g)$ is equal to the number of orbits under the action of $G$. Since $G$ is transitive, the sum is $1$.