# Sundry Coset Results

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## Contents

- 1 Theorems
- 1.1 Element in Coset iff Product with Inverse in Subgroup
- 1.2 Cosets are Equal iff Product with Inverse in Subgroup
- 1.3 Cosets are Equal iff Element in Other Coset
- 1.4 Coset Equals Subgroup iff Element in Subgroup
- 1.5 Elements in Same Coset iff Product with Inverse in Subgroup
- 1.6 Regular Representation on Subgroup is Bijection to Coset

## Theorems

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

Let $x, y \in G$.

Let:

- $x H$ denote the left coset of $H$ by $x$;
- $H y$ denote the right coset of $H$ by $y$.

Then the following results apply:

### Element in Coset iff Product with Inverse in Subgroup

#### Element in Left Coset iff Product with Inverse in Subgroup

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

Then:

- $x \in y H \iff x^{-1} y \in H$

#### Element in Right Coset iff Product with Inverse in Subgroup

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

Then:

- $x \in H y \iff x y^{-1} \in H$

### Cosets are Equal iff Product with Inverse in Subgroup

#### Left Cosets are Equal iff Product with Inverse in Subgroup

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

Then:

- $x H = y H \iff x^{-1} y \in H$

#### Right Cosets are Equal iff Product with Inverse in Subgroup

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

Then:

- $H x = H y \iff x y^{-1} \in H$

### Cosets are Equal iff Element in Other Coset

#### Left Cosets are Equal iff Element in Other Left Coset

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

Then:

- $x H = y H \iff x \in y H$

#### Right Cosets are Equal iff Element in Other Right Coset

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

Then:

- $H x = H y \iff x \in H y$

### Coset Equals Subgroup iff Element in Subgroup

#### Left Coset Equals Subgroup iff Element in Subgroup

- $x H = H \iff x \in H$

#### Right Coset Equals Subgroup iff Element in Subgroup

- $H x = H \iff x \in H$

### Elements in Same Coset iff Product with Inverse in Subgroup

#### Elements in Same Left Coset iff Product with Inverse in Subgroup

- $x, y$ are in the same left coset of $H$ if and only if $x^{-1} y \in H$.

#### Elements in Same Right Coset iff Product with Inverse in Subgroup

- $x, y$ are in the same right coset of $H$ if and only if $x y^{-1} \in H$

### Regular Representation on Subgroup is Bijection to Coset

- 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 $H x$.