Orbit of Element of Group Acting on Itself is Group

Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $*$ be the group action of $\struct {G, \circ}$ on itself by the rule:

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

Then the orbit of an element $x \in G$ is given by:

$\Orb x = G$

Proof

Let $y \in G$.

Then:

\(\ds \exists g \in G: \, \)

\(\ds y\)

\(=\)

\(\ds g \circ x\)

Group has Latin Square Property

\(\ds \leadsto \ \ \)

\(\ds y\)

\(\in\)

\(\ds \Orb x\)

Definition of Orbit

Hence the result.

$\blacksquare$

Also see

• Group Acts on Itself
• Stabilizer of Element of Group Acting on Itself is Trivial

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.15$