Elements in Same Coset 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 $x, y \in G$.


Elements in Same Left Coset iff Product with Inverse in Subgroup

$x, y$ are in the same left coset of $H$ if and only if $x^{-1} y \in H$.


Elements in Same Right Coset iff Product with Inverse in Subgroup

$x, y$ are in the same right coset of $H$ if and only if $x y^{-1} \in H$