I've been reading Keith Conrad's notes on transitive group actions. In them, he states the following theorems:
Theorem 3.2. Let $G$ be a group acting transitively on $X.$ Then $|X|$ divides $|G|.$
Theorem 3.4. Suppose $G$ acts on two finite sets $X$ and $Y$ and there is a function $f : X → Y$ that respects the $G$-actions: $f (gx) = gf (x)$ for all $g ∈ G$ and $x ∈ X.$ If the action is transitive on Y then $|Y | | |X|.$
After proving these statements, he then remarks:
As an exercise, show Theorem 3.2 is a special case of Theorem 3.4.
I'd like some help cleaning up a proof:
My attempt: If $G$ acts transitively on $X$, this is equivalent to the action of $G$ by multiplication on a coset space $G/H$, where $H=G_x$ is the stabilizer of some arbitrary element $x\in X.$ The map $f:G/H\to X, gH\mapsto gx$ is $G$-equivariant, and by Theorem 3.4 $|X|$ divides $|G/H|$ and therefore $|G|.$
Is this proof correct? How it be improved, if so? Are there alternative ways of deriving this implication?