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

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

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $		$\displaystyle x \in y H$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle \exists h' \in H: x = y h'$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Left Coset
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle \exists h = h'^{-1} \in H: x h = y$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Group element properties
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle \exists h \in H: h = x^{-1} y$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Group element properties
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle x^{-1} y \in H$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Left Coset

$\blacksquare$

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

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $		$\displaystyle x \in H y$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle \exists h \in H: x = h y$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Right Coset
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle \exists h \in H: x y^{-1} = h$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Division Laws for Groups
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\iff$	$\displaystyle x y^{-1} \in H$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Right Coset

$\blacksquare$

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

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle x \in y H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle \exists h' \in H: x = y h'\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Left Coset
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle \exists h = h'^{-1} \in H: x h = y\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Group element properties
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle \exists h \in H: h = x^{-1} y\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Group element properties
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle x^{-1} y \in H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Left Coset

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle x \in H y\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle \exists h \in H: x = h y\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Right Coset
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle \exists h \in H: x y^{-1} = h\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Division Laws for Groups
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\iff\)	\(\displaystyle x y^{-1} \in H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Right Coset