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 all $x \in S$:
 * $\Orb x = S$

where $\Orb x$ denotes the orbit of $x \in S$ under $*$.