Cosets in Abelian Group

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Theorem

Let $G$ be an abelian group.

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 is left or right.


Proof

\(\displaystyle \) \(\) \(\displaystyle \forall x, y \in G: x^{-1} y = y x^{-1}\)
\(\displaystyle \) \(\leadsto\) \(\displaystyle \paren {x \equiv^l y \pmod H \iff y \equiv^r x \pmod H}\) Definition of Congruence Modulo Subgroup
\(\displaystyle \) \(\leadsto\) \(\displaystyle \paren {x \equiv^l y \pmod H \iff x \equiv^r y \pmod H}\) Congruence Modulo Subgroup is Equivalence Relation, therefore Symmetric

$\blacksquare$


Sources