Coset of Trivial Subgroup is Singleton
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Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $E := \struct {\set e, \circ}$ denote the trivial subgroup of $\struct {G, \circ}$.
Let $g \in G$.
Then the left coset and right coset of $E$ by $g$ is $\set g$.
Proof
\(\ds g \circ \set e\) | \(=\) | \(\ds \set {g \circ x: x \in \set e}\) | Definition of Left Coset | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {g \circ e}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set g\) |
Similarly:
\(\ds \set e \circ g\) | \(=\) | \(\ds \set {x \circ g: x \in \set e}\) | Definition of Left Coset | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {e \circ g}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set g\) |
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $6$: Cosets: Exercise $2$