Left Coset Equals Subgroup iff Element in Subgroup

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 $x$.

Then:
 * $x H = H \iff x \in H$

Also see

 * Right Coset Equals Subgroup iff Element in Subgroup