Sundry Coset Results

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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

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 $H x$.