Coset Equals Subgroup iff Element in Subgroup
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Theorem
Let $G$ be a group whose identity is $e$.
Let $H$ be a subgroup of $G$.
Let $x \in G$.
Let:
- $x H$ denote the left coset of $H$ by $x$
- $H x$ denote the right coset of $H$ by $x$.
Left Coset Equals Subgroup iff Element in Subgroup
- $x H = H \iff x \in H$
Right Coset Equals Subgroup iff Element in Subgroup
- $H x = H \iff x \in H$