Coset Equals Subgroup iff Element in Subgroup

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Theorem

Let $G$ be a group whose identity is $e$.

Let $H$ be a subgroup of $G$.

Let $x, y \in G$.

Let:

Then:

$(1):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x H = H$	$\iff$	$\displaystyle x \in H$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$(2):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle H x = H$	$\iff$	$\displaystyle x \in H$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$(1): \quad x H = H \iff x \in H$:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x H$	$=$	$\displaystyle H$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x H$	$=$	$\displaystyle e H$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Coset by Identity: $e H = H$
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x e^{-1}$	$\in$	$\displaystyle H$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Equal Cosets iff Product with Inverse in Coset
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x$	$\in$	$\displaystyle H$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Group properties

$\blacksquare$

$(2): \quad H x = H \iff x \in H$ is proved similarly:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle H x$	$=$	$\displaystyle H$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle H x$	$=$	$\displaystyle H e$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Coset by Identity: $H = H e$
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x e^{-1}$	$\in$	$\displaystyle H$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Equal Cosets iff Product with Inverse in Coset
$\displaystyle $	$\displaystyle \iff$	$\displaystyle $	$\displaystyle x$	$\in$	$\displaystyle H$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Group properties

$\blacksquare$

\((1):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x H = H\)	\(\iff\)	\(\displaystyle x \in H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\((2):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle H x = H\)	\(\iff\)	\(\displaystyle x \in H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x H\)	\(=\)	\(\displaystyle H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x H\)	\(=\)	\(\displaystyle e H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Coset by Identity: $e H = H$
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x e^{-1}\)	\(\in\)	\(\displaystyle H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Equal Cosets iff Product with Inverse in Coset
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x\)	\(\in\)	\(\displaystyle H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Group properties

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle H x\)	\(=\)	\(\displaystyle H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle H x\)	\(=\)	\(\displaystyle H e\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Coset by Identity: $H = H e$
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x e^{-1}\)	\(\in\)	\(\displaystyle H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Equal Cosets iff Product with Inverse in Coset
\(\displaystyle \)	\(\displaystyle \iff\)	\(\displaystyle \)	\(\displaystyle x\)	\(\in\)	\(\displaystyle H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Group properties