Obviously, being a professional group theorist, I know what the orbit-stabilizer theorem is. Or at least I thought I did.
I thought that the orbit-stabilizer theorem was that if $G$ is a finite group acting on a set $X$, with stabilizer $H$ of a point $x$, then $|G|$ is the product of $|H|$ and the length of the $G$-orbit of $x$.
But now I see another version of the orbit-stabilizer theorem, which is that if $G$ acts transitively on a set $X$, and $H$ is the point stabilizer, then the action is equivalent to the action on the cosets of $H$. Applying Lagrange's theorem yields the statement that I think is the orbit-stabilizer theorem. I have no name for this statement about transitive actions being equivalent to coset actions.
So my question is: how widespread is this second version? Wikipedia calls the second one the orbit-stabilizer theorem, but a sample of lecture notes that Google served up all used the first version.
This is important because, in a second course on group theory for example, you might want to use the orbit-stabilizer theorem, and you probably won't go and check the previous lecturer's notes to find out what it is they chose. Named theorems should mean the same to all people, lest we fall down some Wittgensteinian rabbit hole.
The way I state it in my final year group theory course is:
Let $G$ act on $\Omega$ and let $\alpha \in \Omega$. Then there is a bijection between the right cosets $G_\alpha g$ of $G_\alpha$ in $G$ and $\alpha^G$ defined by $G_\alpha g \mapsto \alpha^g$. In particular, if $G$ is finite then $|G|=|\alpha ^G| |G_\alpha |$.
That has the advantage of giving more information than the "in particular" part, which is often useful, and also it does not restrict the statement to finite groups. I state the equivalence of transitive actions with coset actions as a separate result.