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.

For any $x \in S$, let:
 * $\operatorname{Orb} \left({x}\right) = S$

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

Then $G$ acts transitively on $S$.

$*$ is known as a transitive group action.