Orbit-Stabilizer theorem

Definition A group action of a group on a set is just an operation $g s \in S$ for $s \in S$.

Corollary This gives an an equivalence relation on the set: $x \sim y$ if $exists g, y = gx$.

Definition The orbit ($Orb(s)$) of an element $s$ of the equivalence class containing $s$ i.e. the set is all the elements $\sim$ to $s$. Alternatively it's the canonical map $S \to S/\sim$.

Definition A group acts transitively if there is only one orbit (every element is related to every other).

Definition The stabalizer ($Stab(s))$ is the set of group elements $g$ such that $gs = s$.

Theorem $Stab(s) \le G$ (meaning it's a subgroup).


Theorem $$|Orb(x)|\cdot|Stab(s)|=|G|.$$
proof: It's also easy to see this identity holds if either of the factors are 1. First, Let $y = gx$ and $g' x = x$ then $g'^g y = g^{-1}g'gy = y$. This tells us that $|Stab(x)| = |Stab(y)|$ when $y = gx$ (because the group automorphism gives us a bijection between the elements which stabilize x and those which stabilize y), and there are |Orb(x)| such relations $x$,$y$,$\ldots$ so we have the theorem.