Left 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 $x H$ denote the left coset of $H$ by $x$.


Then:

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


Proof

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

$\blacksquare$


Also see


Sources