Regular Representation on Subgroup is Bijection to Coset/Left

Theorem

Let $G$ be a group.

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

Let $x, y \in G$.

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

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 $x H$.

Proof

Let $h \in H$.

Then:

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

Thus:

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

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.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem