Regular Representation on Subgroup is Bijection to Coset
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Theorem
Let $G$ be a group.
Let $H$ be a subgroup of $G$.
Let $x, y \in G$.
Left Coset
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 $x H$.
Right Coset
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 $H x$.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem