Cosets are Equal iff Element in Other Coset
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Theorem
Let $\struct {G, \circ}$ be a group.
Let $H$ be a subgroup of $G$.
Let $x, y \in G$.
Left Cosets are Equal iff Element in Other Left Coset
Let $x H$ denote the left coset of $H$ by $x$.
Then:
- $x H = y H \iff x \in y H$
Right Cosets are Equal iff Element in Other Right Coset
Let $H x$ denote the right coset of $H$ by $x$.
Then:
- $H x = H y \iff x \in H y$