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

\(\ds a * \paren {b * g' H}\)

\(=\)

\(\ds a * \paren {\paren {b g'} H}\)

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 $\text {GA} 1$.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $6$: Cosets: Exercise $9$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 53$