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:


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