Orbit-stabilizer theorem

The Orbit-Stabilizer Theorem links the size of a group's actions on a set to the structure of the group itself. It states that for any element in the set, the size of its orbit (all positions it can reach through the group's actions) multiplied by the size of its stabilizer (subgroup fixing that element) equals the size of the entire group. Essentially, it connects how the group "moves" elements and the subgroup that "keeps" a certain element fixed, providing a fundamental way to understand symmetry and group behavior in various mathematical contexts.