Conjugacy Action on Abelian Group is Trivial
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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: $\text{(v)}$