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

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

$\blacksquare$


Also see


Sources