Group Action on Coset Space is Transitive

Theorem
Let $G$ be a group whose identity is $e$.

Let $H$ be a subgroup of $G$.

Let $*: G \times G / H \to G / H$ be the action on the (left) coset space:
 * $\forall g \in G, \forall g' H \in G / H: g * \paren {g' H} := \paren {g g'} H$

Then $G$ is a transitive group action.

Proof
It is established in Action of Group on Coset Space is Group Action that $*$ is a group action.

It remains to be shown that:
 * $\forall g' H \in G / H: \Orb {g' H} = G / H$

where $\Orb {g' H}$ denotes the orbit of $g' H \in G / H$ under $*$.

Let $a H, b H \in G / H$ such that $a H \ne b H$.

We have that:
 * $\exists x \in G: x a = b$

and so:
 * $x * a H = \paren {x a} H = b H$

and so:
 * $b H \in \Orb {a H}$

As both $a$ and $b$ are arbitrary, the result follows.