$G$ is a group acting on $S$. $H$ is a subgroup of $G$ and let $s$ be anything in $S$. Is it true that the size of the orbit of $s$ under action of $G$ is divisible by the size of the orbit of $s$ under action of $H$?
2026-04-06 04:36:14.1775450174
Orbit of a subgroup
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If G is the symmetric group on n letters, then the orbit of s has exactly n elements. Saying that H ≤ G means that H is a permutation group on n letters. Does every permutation group on n letters have orbits that are divisors of n?
Perhaps check n = 3 to start. A similar investigation n = 5 should also be instructive.