Mathematics
Permutation Groups and the Orbit-Stabilizer Theorem
Quick fact
The orbit-stabilizer theorem states that for a group G acting on a set X, for any element x, |G| = |Orbit(x)| × |Stabilizer(x)|. This simple formula lets us compute the size of enormous geometric symmetry groups without listing them.
Why this is interesting
Think of a Rubik's cube. Every twist rearranges the stickers—that's a permutation. But did you know that the number of possible arrangements you can reach is exactly the size of a certain group? The orbit-stabilizer theorem is the key that counts them.