The Orbit-Stabilizer says that, given a group G which acts on a set X, then there exists a bijection between the orbit of an element x∈X and the set of left cosets of the stabilizer group of x in G. In other words, that the cardinality of the orbit of an element x∈X is equal to the index of its stabilizer subgroup in G.

I’ve seen two different texts present this, both of which explicitly say that this captures a very intuitive idea. I’m sorry if it’s obvious, but I don’t see the intuition behind this.

I’ve asked a few questions looking for intuition now, and have received outstanding advice. As such, again I’m looking to the community to share some of their insights on this idea, and how they think of this theorem. As always, any help is greatly appreciated. Thanks!

This is not my answer, but Gowers explains the orbit-stablizer theorem in an excellent way.

Think about the symmetric group of a cube. Call this group G

and it acts on the cube. If you want to know how many elements there are in G

, then you can think about it in two steps.

Step I: If you fix one face, there are 4 ways to move the cube because you can only rotate the cube now. (These are the stabilizers )

Step II: There are six possible choice where this face can go. (Orbit of the face).

So you figure out |G|=4⋅6

. That is the intuition.