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 $x\in X$ 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 $x\in X$ 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!
The statement of the theorem that I know says that if we partition $G$ into cosets of some $G_x,$ then we get an isomorphism of $G$-sets $G/G_x\to O(x),aG_x\mapsto a\cdot x.$
Here is my naïve way of thinking about it. Given a $G$-space $X$ and finite $G,$ the orbit-stabilizer theorem implies the "counting theorem" $|G_x||O(x)|=|G|.$ Since the size of $G$ is fixed, this is telling us that the larger the orbit is, the fewer elements of $G$ can possibly fix $x,$ while the smaller the orbit is, the more elements of $G$ must fix $x.$ This seems pretty intuitive, but it's not obvious. For example, we might think it possible that $|O(x)|$ is small, while many elements of $G$ act nontrivially but in the same way on $x.$ The orbit-stabilizer theorem disallows this, letting us determine the size of the stabilizer from the size of the corresponding orbit, and vice-versa.