# Right Coset by Identity

## Theorem

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

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

Then:

$H = H e$

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

## Proof

We have:

 $\displaystyle H e$ $=$ $\displaystyle \set {x \in G: \exists h \in H: x = h e}$ Definition of Right Coset of $H$ by $e$ $\displaystyle$ $=$ $\displaystyle \set {x \in G: \exists h \in H: x = h}$ Definition of Identity Element $\displaystyle$ $=$ $\displaystyle \set {x \in G: x \in H}$ $\displaystyle$ $=$ $\displaystyle H$

So $H = H e$.

$\blacksquare$

## Also see

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

$H e = H \set e$