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 $$\rho_x$$.