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 $e$.

We have:

$\ds H e$

$=$

$\ds \set {x \in G: \exists h \in H: x = h e}$

Definition of Right Coset of $H$ by $e$

$\ds $

$=$

$\ds \set {x \in G: \exists h \in H: x = h}$

Definition of Identity Element

$\ds $

$=$

$\ds \set {x \in G: x \in H}$

$\ds $

$=$

$\ds H$

So $H = H e$.

$\blacksquare$

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

$H e = H \set e$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.1$. The quotient sets of a subgroup: Example $112$
1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$: Example $30$
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions