This is part of a homework problem for a graduate course on abstract algebra.
Given a transitive G-set $X$, show that the map that assigns to $x \in X$ its stabilizer defines a morphism of G-sets with G acting on its set of subgroups by conjugation.
I'm very much confused by this.
So we have a transitive G-set $X$, which means (among other things) every orbit is equal to all of $X$ (alternatively, it has only one orbit).
We also have a map $f: X \rightarrow S$ given by the rule $f(x) = stab_G(x)$, where $S$ is the set of subgroups of $G$.
A morphism of G-sets would be a map $h: G_1 \rightarrow G_2$ such that $gh(x) = h(gx)$ for all $g \in G$.
But part of the specification of this morphism is that G is taken to act on its subgroups by conjugation.
I'm completely confused by this. I don't see how to use our function on a fixed G-set to get a function between arbitrary G-sets (or if this is even what's expected), I don't see how the action of the G-set (by conjugation over subgroups) falls into place, and I'm a bit confused by the change in the action from when we specify our $X$ (by specifying that it's a G-set, we must already have an action) to the new action by conjugation.
I also realize that this would be more appropriate of a question for my prof/TA/classmates. I was planning to ask on Monday but forgot about the statutory holiday (it's due Tuesday morning). I'm not seeking a solution, just help in understanding the question.