Right Cosets are Equal iff Product with Inverse in Subgroup

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Theorem

Let $G$ be a group.

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

Let $x, y \in G$.

Let $H x$ denote the right coset of $H$ by $x$.


Then:

$H x = H y \iff x y^{-1} \in H$


Proof

\(\ds H x\) \(=\) \(\ds H y\)
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(\equiv^r\) \(\ds y \bmod H\) Right Coset Space forms Partition
\(\ds \leadstoandfrom \ \ \) \(\ds x y^{-1}\) \(\in\) \(\ds H\) Equivalent Statements for Congruence Modulo Subgroup

$\blacksquare$


Also see


Sources

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