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