# 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.

$\blacksquare$