Action of Group on Coset Space is Group Action

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

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

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

Then:

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

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

$\blacksquare$