Orbit of Element of Group Acting on Itself is Group

From ProofWiki
Jump to navigation Jump to search


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:

$\Orb x = G$


Let $y \in G$.


\(\ds \exists g \in G: \, \) \(\ds y\) \(=\) \(\ds g \circ x\) Group has Latin Square Property
\(\ds \leadsto \ \ \) \(\ds y\) \(\in\) \(\ds \Orb x\) Definition of Orbit

Hence the result.


Also see