Transitive Action - Imported Knowlpedia Content

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 $g\cdot x = y$ .

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).

Equivalent characterizations

Equivalently, the action has exactly one orbit.

Remarks

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