Conjugation
From Maths
Contents
[<hidetoc>]Definition
Two elements g,h of a group (G,×) are conjugate if:
- ∃x∈G[xgx−1=h]
Conjugation operation
Let x in G be given, define:
- Cx:G→G as the automorphism (recall that means an isomorphism of a group onto itself) which:
- g↦xgx−1
This association of x↦cx is a homomorphism of the form G→Aut(G) (or indeed G→(G→G) instead)
This operation on G is called conjugation[1]
TODO: Link with language - "the conjugation of x is the image of cx" and so forth
Proof of clams
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Claim: The map Cx:G→G given by g↦xgx−1 is an automorphism
[Expand]
Claim: The family {Cx|x∈G} form a group, and x↦cx is a homomorphism from G to this family
See also
References
- <cite_references_link_accessibility_label> ↑ Algebra - Serge Lang - Revised Third Edition - GTM