Transitive Action

A group action of a group $G$ on a set $X$ is transitive if for all $x,y\in X$ there exists $g\in G$ such that $g\cdot x = y$. Equivalently, the action has exactly one orbit .

Transitive actions are the natural context for coset actions: if $H\le G$, then the action of $G$ on $G/H$ is always transitive.

Examples:

• The natural action of $S_n$ on $\{1,\dots,n\}$ is transitive.
• The action of $G$ on $G/H$ by left multiplication is transitive for any subgroup $H$.
• The conjugation action of a nonabelian group on itself is generally not transitive (it has multiple conjugacy classes).