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:

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

\(\ds g * h\)

\(=\)

\(\ds g \circ h \circ g^{-1}\)

\(\ds \)

\(=\)

\(\ds h \circ g \circ g^{-1}\)

Definition of Abelian Group: $g$ commutes with $h$

\(\ds \)

\(=\)

\(\ds h \circ e\)

Group Axiom $\text G 3$: Existence of Inverse Element

\(\ds \)

\(=\)

\(\ds h\)

Group Axiom $\text G 2$: Existence of Identity Element

Thus by definition of orbit:

$\Orb h = \set h$

Hence the result by definition of trivial group action.

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions: Examples of group actions: $\text{(v)}$