Conjugation

Definition

Two elements [ilmath]g,h[/ilmath] of a group [ilmath](G,\times)[/ilmath] are conjugate if:

• [ilmath]\exists x\in G[xgx^{-1}=h][/ilmath]

Conjugation operation

Let [ilmath]x[/ilmath] in [ilmath]G[/ilmath] be given, define:

• [ilmath]C_x:G\rightarrow G[/ilmath] as the automorphism (recall that means an isomorphism of a group onto itself) which:

• [ilmath]g\mapsto xgx^{-1} [/ilmath]

This association of [ilmath]x\mapsto c_x[/ilmath] is a homomorphism of the form [ilmath]G\rightarrow\text{Aut}(G)[/ilmath] (or indeed [ilmath]G\rightarrow(G\rightarrow G)[/ilmath] instead)

This operation on [ilmath]G[/ilmath] is called conjugation

---

TODO: Link with language - "the conjugation of x is the image of [ilmath]c_x[/ilmath]" and so forth

---

Proof of clams

Claim: The map [ilmath]C_x:G\rightarrow G[/ilmath] given by [ilmath]g\mapsto xgx^{-1} [/ilmath] is an automorphism

---

---

TODO: On note-paper

---

Claim: The family [ilmath]\{C_x\vert x\in G\} [/ilmath] form a group, and [ilmath]x\mapsto c_x[/ilmath] is a homomorphism from [ilmath]G[/ilmath] to this family

---

---

TODO: Not done yet

---

See also

• Commutator
• Group