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 $x$

$H x$ 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

$H x = H \iff x \in H$