Action of Group on Coset Space is Group Action

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Theorem

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

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


Let $*: G \times G / H \to G / H$ be the action on the (left) coset space:

$\forall g \in G, \forall g' H \in G / H: g * \paren {g' H} := \paren {g g'} H$

Then $G$ is a group action.


Proof

\(\ds a * \paren {b * g' H}\) \(=\) \(\ds a * \paren {\paren {b g'} H}\) Definition of $*$
\(\ds \) \(=\) \(\ds \paren {a \paren {b g'} } H\) Definition of $*$
\(\ds \) \(=\) \(\ds \paren {a b} g' H\) Group Axiom $\text G 1$: Associativity

demonstrating that $*$ satisfies Group Action Axiom $\text {GA} 2$.


Then:

\(\ds e * g' H\) \(=\) \(\ds \paren {e g'} H\) Definition of $*$
\(\ds \) \(=\) \(\ds g'H\) Group Axiom $\text G 2$: Existence of Identity Element

demonstrating that $*$ satisfies Group Action Axiom $\text {GA} 1$.

$\blacksquare$


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