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