I'm currently working through the proof of Theorem 1, III.12 in Mumford's "Abelian Varieties".
Let $G$ be a finite $k$-group scheme acting on an affine $k$-scheme $X:=\text{Spec}~A$ and let $\phi: G\times_{\text{k}} X\rightarrow X\times_{\text{k}} X$ be the free action given by $(g,x)\mapsto (gx,x)$. The map $\phi$ is then a closed immersion.
Why does then $\phi$ factor through a closed immersion $G\times_k X\rightarrow X\times_{X/G}X$?
Thank you very much in advance!
Since $gx$ and $x$ lie in the same orbit, $\phi$ factors through $X \times_{X/G} X$. The morphism $G \times X \to X \times_{X/G} X$ is a closed immersion, since $X \times_{X/G} X \to X \times X$ is a closed immersion and $\phi$ is a closed immersion. Is this enough?