What can we say about the sets X on which a group G acts freely and transitively?

Question

The "philosophy" of (group) actions is such that the most rigid conditions we can think of (abstractly) are freeness and transitivity.

Let me introduce some terminology to make this idea clearer. For $G$ a group, we call a set $X$ a rigid set for $G$ if there exists a group action $. :G \times X \rightarrow X$ that is both free and transitive.

Does any group $G$ admits a rigid set $X$? What can we say about the category of rigid sets for $G$ where morphisms are $G$-equivariant maps?

PS: this question is motivated by the fact that in practice, most natural actions aren't free but only faithful. For example, the natural action of $SO(3)$ on $\mathbb{R}^3$ is faithful and transitive, but certainly not free. Yet, it seems to me that the "good object" on which a group $G$ should act has to be a rigid set for $G$. Does a rigid set for $SO(3)$ is known to exist ?

Answer 1

Every free and transitive $G$-set is isomorphic to $G$ acting on itself by left multiplication. Every morphism of $G$-sets between free and transitive $G$-sets is an isomorphism. So the category is equivalent to the category with one object with automorphism group $G$.

A natural example of a space on which $SO(3)$ acts freely and transitively is the space of oriented orthonormal bases of $\mathbb{R}^3$.

These questions become substantially more interesting in a setting other than sets. See torsor and principal bundle for details.