Left Regular Representation is Transitive Group Action

Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $*: G \times G \to G$ be the group action:
 * $\forall g, h \in G: g * h = \map {\lambda_g} h$

where $\lambda_g$ is the left regular representation of $G$ with respect to $g$.

Then $*$ is a transitive group action.

Proof
Let $g, h \in G$.

Then:

$h$ is arbitrary, therefore the above holds for all $h \in G$.

By definition of orbit:
 * $\Orb h = \set {t \in G: \exists g \in G: g * h = t}$

That is:
 * $\Orb h = G$

Hence the result by definition of transitive group action.

Also see

 * Right Regular Representation by Inverse is Transitive Group Action