Cosets in Abelian Group

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Theorem

Then every right coset modulo $H$ is a left coset modulo $H$.

That is:

$\forall x \in G: x H = H x$

In an abelian group, therefore, we can talk about congruence modulo $H$ and not worry about whether it's left or right.

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $		$\displaystyle \forall x, y \in G: x^{-1} y = y x^{-1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle \left({x \ \equiv^l \ y \ \left({\bmod H}\right) \iff y \ \equiv^r \ x \ \left({\bmod H}\right)}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Congruence Modulo a Subgroup
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle \left({x \ \equiv^l \ y \ \left({\bmod H}\right) \iff x \ \equiv^r \ y \ \left({\bmod H}\right)}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Congruence Modulo a Subgroup is an Equivalence, therefore Symmetric

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle \forall x, y \in G: x^{-1} y = y x^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left({x \ \equiv^l \ y \ \left({\bmod H}\right) \iff y \ \equiv^r \ x \ \left({\bmod H}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Congruence Modulo a Subgroup
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left({x \ \equiv^l \ y \ \left({\bmod H}\right) \iff x \ \equiv^r \ y \ \left({\bmod H}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Congruence Modulo a Subgroup is an Equivalence, therefore Symmetric