Conjugacy Action is not Transitive

Theorem

Let $\struct {G, \circ}$ be a non-trivial 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 not a transitive group action.

For $G$ to be a transitive group action, the orbit of any element of $G$ needs to be the whole of $G$.

Take $h = e$.

Then:

$\ds \forall g \in G: \, $

$\ds g * e$

$=$

$\ds g \circ e \circ g^{-1}$

Definition of Conjugacy Action

$\ds $

$=$

$\ds g \circ g^{-1}$

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

$\ds $

$=$

$\ds e$

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

Thus by definition of orbit:

$\Orb e = \set e$

Only when $G$ is the trivial group, that is, when:

$G = \set e$

then:

$\Orb e = G$.

Hence the result by definition of transitive group action.

$\blacksquare$

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