Coset by Identity

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

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

Then:
 * $e H = H = H e$

where:
 * $e H$ is the left coset of $H$ by $e$
 * $H e$ is the right coset of $H$ by $e$.

Proof

 * $e H = H$:

We have:

So $e H = H$.


 * $H = H e$:

Similarly, we have:

So $H = H e$.

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


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