Transitive action

Definition

Let a group $G$ act on a nonempty set $X$. The action is transitive when for all $x, y \in X$ there is a $g \in G$ with $g \cdot x = y$.

Remarks

The action is transitive if and only if it has exactly one orbit. The action of $S_n$ on $\{1, \dots, n\}$ is transitive. The action of $G$ on the left cosets of a subgroup $H$ is transitive. Each transitive action of $G$ is isomorphic to such an action.

Explanations

L1 human

A transitive action can move any point to any other point. So the set does not split into separate pieces under the action. The rotations of a square act transitively on its four corners.