# Coset by Identity

## Theorem

Let $G$ be a group whose identity is $e$.

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

### Left Coset by Identity

Then:

$e H = H$

where $e H$ is the left coset of $H$ by $e$.

### Right Coset by Identity

Then:

$H = H e$

where $H e$ is the right coset of $H$ by $e$.

## Also see

This is consistent with the definition of the concept of coset by means of the subset product:

$e H = \left\{{e}\right\} H$
$H e = H \left\{{e}\right\}$