What is the relationship between the orbit-stabilizer theorem and Lagrange's theorem?

Question

Is Lagrange's theorem used to prove that the length of the orbit times the order of the stabilizer is the order of the group, or is Lagrange's theorem a corollary of the orbit-stabilizer theorem?

Answer 1

Usually, Lagrange's theorem is used to prove the orbit-stabilizer theorem, not the other way around. See the following proof from "Abstract Algebra: Theory and Applications":

However, if someone could figure out how to prove the orbit-stabilizer theorem without using Lagrange's Theorem, then you could prove Lagrange's Theorem as a corollary of the orbit-stabilizer theorem, as Tsemo Aristide showed.

Answer 2

Lagrange is a corollary, if $H$ is a subgroup of $G$, $H$ acts on $G$ by left multiplication, the orbit of $1$ is $H$ so $|H|Or(G/H)=|G|$ where $Or(G/H)$ is the cardinal of the orbit space.