Equal Cosets iff Product with Inverse in Coset

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Theorem

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

Let $x, y \in G$.

Let:

Then:

$(1):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x H = y H$	$\iff$	$\displaystyle x^{-1} y \in H$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$(2):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle H x = H y$	$\iff$	$\displaystyle x y^{-1} \in H$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$(1): \quad x H = y H \iff x^{-1} y \in H$:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $		$\displaystyle x H = y H$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle x \ \equiv^l \ y \ \left({\bmod H}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Congruence Class Modulo Subgroup is Coset
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle x^{-1} y \in H$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Equivalent Statements for Congruence Modulo a Subgroup

$\blacksquare$

$(2): \quad H x = H y \iff x y^{-1} \in H$:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $		$\displaystyle H x = H y$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle x \ \equiv^r \ y \ \left({\bmod H}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Congruence Class Modulo Subgroup is Coset
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle x y^{-1} \in H$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Equivalent Statements for Congruence Modulo a Subgroup

$\blacksquare$

\((1):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x H = y H\)	\(\iff\)	\(\displaystyle x^{-1} y \in H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\((2):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle H x = H y\)	\(\iff\)	\(\displaystyle x y^{-1} \in H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle x H = y H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle x \ \equiv^l \ y \ \left({\bmod H}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Congruence Class Modulo Subgroup is Coset
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle x^{-1} y \in H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Equivalent Statements for Congruence Modulo a Subgroup

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle H x = H y\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle x \ \equiv^r \ y \ \left({\bmod H}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Congruence Class Modulo Subgroup is Coset
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle x y^{-1} \in H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Equivalent Statements for Congruence Modulo a Subgroup