Regular Representation on Subgroup is Bijection to Coset

Theorem
Let $G$ be a group and let $H$ be a subgroup of $G$.

Let $x, y \in G$.

Let:
 * $y H$ denote the left coset of $H$ by $y$;
 * $H y$ denote the right coset of $H$ by $y$.

Then:
 * The mapping $\lambda_x: H \to x H$, where $\lambda_x$ is the left regular representation of $H$ with respect to $x$, is a bijection from $H$ to $x H$.


 * 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
Follows from Regular Representations in Group are Permutations.

Let $h \in H$.

Then:
 * $\lambda_x \left({h}\right) = x h \in x H$

Thus:
 * $\forall h \in H: \lambda_x h \in x H$

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

A permutation is a bijection by definition,

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

Exactly the same argument applies to $\rho_x$.