Conjugacy Action on Abelian Group is Trivial

Theorem

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

Let $*: G \times G \to G$ be the conjugacy group action:

$\forall g, h \in G: g * h = g \circ h \circ g^{-1}$

Then $*$ is a trivial group action.

Proof

For $G$ to be a trivial group action, the orbit of any element of $G$ is a singleton containing only that element.

Take $h \in G$.

Then:

 $\displaystyle \forall g \in G: \ \$ $\displaystyle g * h$ $=$ $\displaystyle g \circ h \circ g^{-1}$ $\displaystyle$ $=$ $\displaystyle h \circ g \circ g^{-1}$ Definition of Abelian Group: $g$ commutes with $h$ $\displaystyle$ $=$ $\displaystyle h \circ e$ Group axiom $G3$: properties of inverse element $\displaystyle$ $=$ $\displaystyle h$ Group axiom $G2$: properties of identity element

Thus by definition of orbit:

$\Orb h = \set h$

Hence the result by definition of trivial group action.

$\blacksquare$