# Element in Coset iff Product with Inverse in Subgroup

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## Theorem

Let $\struct {G, \circ}$ be a group.

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

Let $x, y \in G$.

#### Element in Left Coset iff Product with Inverse in Subgroup

Let $y H$ denote the left coset of $H$ by $y$.

Then:

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

#### Element in Right Coset iff Product with Inverse in Subgroup

Let $H \circ y$ denote the right coset of $H$ by $y$.

Then:

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