Degree of finite group schemes

Let $\pi: G \rightarrow S$ be a finite flat group scheme over a locally noetherian connected base scheme $S$. Its degree is defined as the rank of the locally free $\mathcal O_S$-module $\pi_* \mathcal O_G$. Let $H \subset G$ be a closed subgroup scheme of $G$ which is also finite flat over $S$.

I want to show that the degree of $H$ divides the degree of $G$. How does one do this? I guess this must be easy but I'm somehow stuck.

• The proof should be the same as for Lagrange's theorem: $H$ acts freely on $G$ by translations, making $G$ an $H$-torsor over $G/H$. – Keerthi Madapusi Pera May 18 '12 at 10:28
• Maybe you want to assume your $\pi$ is finite locally free instead of just finite flat, if your $S$ might not be locally Noetherian. I feel like $\pi_*\mathscr{O}_G$ might not be locally free if $\pi$ is not locally of finite presentation. – Keenan Kidwell May 18 '12 at 16:24
• @Keenan: that's a good point, I simply forgot the locally noetherian assumption. – Veen May 18 '12 at 17:09

This can be seen from the existence of the quotient $G/H$ as a finite flat $S$-scheme (and an $H$-torsor). One shows that the natural map $G \rightarrow G/H$ is finite flat of order $[H : S]$; the conclusion then follows from the product formula $[G : S] = [G : G/H] [G/H : S]$. Let me give you a reference where all this is spelled out in detail (and which I am basically copying):