Regular Representation on Subgroup is Bijection to Coset/Right

Theorem

Let $G$ be a group.

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

Let $x, y \in G$.

Let $H y$ denote the right coset of $H$ by $y$.

The mapping $\rho_x: H \to H x$, where $\rho_x$ is the right regular representation of $H$ with respect to $x$, is a bijection from $H$ to $H x$.

Proof

Let $h \in H$.

Then:

$\map {\rho_x} h = h x \in H x$

Thus:

$\forall h \in H: \map {\rho_x} h \in H x$

demonstrating that $\rho_x: H \to H x$ is a mapping.

A permutation is a bijection by definition.

As Regular Representations in Group are Permutations, it follows that $\rho_x$ is a bijection.

$\blacksquare$