Group Action on Coset Space is Transitive
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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 transitive group action.
Proof
It is established in Action of Group on Coset Space is Group Action that $*$ is a group action.
It remains to be shown that:
- $\forall g' H \in G / H: \Orb {g' H} = G / H$
where $\Orb {g' H}$ denotes the orbit of $g' H \in G / H$ under $*$.
Let $a H, b H \in G / H$ such that $a H \ne b H$.
We have that:
- $\exists x \in G: x a = b$
and so:
- $x * a H = \paren {x a} H = b H$
and so:
- $b H \in \Orb {a H}$
As both $a$ and $b$ are arbitrary, the result follows.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $6$: Cosets: Exercise $9$