Regular Action

A group action of a group $G$ on a set $X$ is regular if it is both free and transitive . Equivalently, for every pair $x,y\in X$ there exists a unique $g\in G$ such that $g\cdot x = y$.

Regular actions identify the set $X$ with the underlying set of $G$ (non-canonically, after choosing a basepoint), and are a standard way to model $G$ as permutations of a set.

Examples:

• The left translation action of $G$ on itself is regular.
• The action of $G$ on $G/H$ is regular iff $H$ is trivial.
• Any group acting on itself by left multiplication provides a regular action, hence a faithful permutation representation.