# Element of Group is in Unique Coset of Subgroup

## Theorem

Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Let $x \in G$.

### Left Coset

There exists a exactly one left coset of $H$ containing $x$, that is: $x H$

### Right Coset

There exists a exactly one right coset of $H$ containing $x$, that is: $H x$