TITLE: What is the relationship between the orbit-stabilizer theorem and Lagrange's theorem?
QUESTION [5 upvotes]: 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?

REPLY [6 votes]: 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.