Definition 6

Let G be a group. An operation of G on a set E is called transitive if there exists an element $x \in E$ whose orbit is E. A G-set E is called homogeneous if the operation of G on E is transitive.

It is also said that G operates transitively on E; or that E is a homogeneous set under G. It amounts to the same to say that E is non-empty and that, for all elements $x$ and $y$ of E, there exists an element $\alpha \in G$ such that $\alpha.x = y$.