Left 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:
- $e H = H$
where $e H$ is the left coset of $H$ by $e$.
Proof
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 $e H = H$.
$\blacksquare$
Also see
This is consistent with the definition of the concept of coset by means of the subset product:
- $e H = \set e H$
Sources
- 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 $1$