Cosets are Equal iff Product with Inverse in Subgroup
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Theorem
Let $G$ be a group and let $H$ be a subgroup of $G$.
Let $x, y \in G$.
Left Cosets are Equal iff Product with Inverse in Subgroup
Let $x H$ denote the left coset of $H$ by $x$.
Then:
- $x H = y H \iff x^{-1} y \in H$
Right Cosets are Equal iff Product with Inverse in Subgroup
Let $H x$ denote the right coset of $H$ by $x$.
Then:
- $H x = H y \iff x y^{-1} \in H$