Given a group of small order (<30), how does one go about systematically finding all the transitive $G$-sets up to isomorphism?
By $X$ and $Y$ being isomorphic we mean there are maps $f:X \rightarrow Y$ and $g:Y \rightarrow X$ preserving the group action.
The main issue I'm having trouble with is that a $G$-set can be any kind of set, and one could in principle get quite creative with how a group $G$ acts on it. The orbit-stabilizer theorem narrows down the orders of such $G$-sets (they must divide the order of $G$), but one could still have a variety of different actions on sets of a given order.
I've also proven the following for transitive $G$-sets $X, Y$:
If $f:X \rightarrow Y$ is a morphism, then the stabilizer for any $x$ is a subgroup of the stabilizer for $f(x)$, and $|Y|$ divides $|X|$.
The map $h: X \rightarrow \{ H \leq G\}$ given the rule $h(x) = Stab_G(x)$ is a morphism on $X$ with $G$ acting on its subgroups via conjugation. Its image is also a transitive $G$-set.
This final fact seems to be the key, in that it might allow one to look solely at conjugates of subgroups of $G$ for all transitive $G$-sets up to isomorphism, but after playing around with it for hours I haven't been able to squeeze anything out of it.
This is used in a problem set for a graduate course on group theory. We need to find the posets (the ordering given by the existence of a morphism) of isomorphism classes of transitive $G$-sets of specific groups, but I'm struggling to get a handle on the $G$-sets.
Suppose $X$ is a transitive $G$-set; I am writing the action of $g \in G$ on $x \in X$ as $x g$.
Fix $x_{0} \in X$, and consider the map $$ \psi: G \to X \qquad g \mapsto x_{0} g. $$ This is surjective, as $X$ is transitive. Moreover, the equivalence relation induced by $\psi$ on $G$ is $$ g \sim h \text{ iff } \psi(g) = \psi(h) \text{ iff } x_{0} g = x_{0} h \text{ iff } g h^{-1} \in S = \operatorname{Stab}_{G}(x_{0}) \text{ iff } S g = S h. $$ So this yields the isomorphism of $G$-sets between the coset space $\{ S g : g \in G \}$ and $X$ given by $S g \mapsto x_{0} g$.