Right Coset by Identity

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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:

$H e = H \set e$



Sources