Difference between revisions of "Conjugation"
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Revision as of 22:21, 11 May 2015
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[1]
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
References
- ↑ Algebra - Serge Lang - Revised Third Edition - GTM
