Left Coset by Identity

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

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


$e H = H$

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 $e H = H$.


Also see

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

$e H = \set e H$