Definition:Transitive Group Action

Definition
Let $G$ be a group.

Let $S$ be a set.

Let $*: G \times S \to S$ be a group action.

The group action is transitive for any $x,y\in S$ there exists $g\in G$ such that $g * x = y$.

That is, for any $x \in S$:
 * $\operatorname{Orb} \left({x}\right) = S$

where $\operatorname{Orb} \left({x}\right)$ denotes the orbit of $x \in S$ under $*$.