Left Coset by Identity

Theorem

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

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

Then:

$e H = H$

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

We have:

$\ds e H$

$=$

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

Definition of Left Coset of $H$ by $e$

$\ds $

$=$

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

Definition of Identity Element

$\ds $

$=$

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

$\ds $

$=$

$\ds H$

So $e H = H$.

$\blacksquare$

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

$e H = \set e H$

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 $30$
1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Definition $5.1$: Remark $1$