Conjugation

Definition

Two elements $g,h$ of a group $(G,\times)$ are conjugate if:

• $\exists x\in G[xgx^{-1}=h]$

Conjugation operation

Let $x$ in $G$ be given, define:

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

• $g\mapsto xgx^{-1}$

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

This operation on $G$ is called conjugation

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TODO: Link with language - "the conjugation of x is the image of $c_x$" and so forth

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Proof of clams

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

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TODO: On note-paper

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

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TODO: Not done yet

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See also

• Commutator
• Group