Orbit of Subgroup Action is Coset

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Theorem

Let $\struct {G, \circ}$ be a group.

Let $\struct {H, \circ}$ be a subgroup of $G$.

Let $*: H \times G \to G$ be the subgroup action defined for all $h \in H, g \in G$ as:

$\forall h \in H, g \in G: h * g := h \circ g$


The orbit of $x \in G$ is the right coset by $x$ of $H$:

$\Orb x = H x$


Proof

From Subgroup Action is Group Action we have that $*$ is a group action.

Let $x \in G$.

Then:

\(\ds \Orb x\) \(=\) \(\ds \set {g \in G: \exists h \in H: g = h * x}\) Definition of Orbit
\(\ds \) \(=\) \(\ds \set {g \in G: \exists h \in H: g = h \circ x}\) Definition of $*$
\(\ds \) \(=\) \(\ds H x\)

Hence the result, by definition of right coset.

$\blacksquare$


Sources

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