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Conjugation is an automorphism
Statement
Conjugation is an automorphism.
For each , the map , , is an automorphism.
Facts & Assumptions
Given: A group and .
An automorphism is a bijective group homomorphism (Group isomorphisms, automorphisms and the set ).
Inverses reverse products and is the inverse of (In a group , and , the order of the last product being essential).
Proof
Associativity gives , so is a homomorphism.
The map is inverse to by cancellation.
Thus is a bijective homomorphism and hence an automorphism.
Depends on
Used by
- There are exactly two isomorphism classes of groups of order 105 Corollary
- The subgroup ⟨(1 2 3),(1 2)(3 4)⟩≤ S₄ has order 12 but no subgroup of order 6, so Cauchy's theorem does not extend to composite divisors Counterexample
- Inner automorphisms and Inn(G) Definition
- The core Core_G(H)=⋂_g∈ GgHg⁻¹ of a subgroup Definition
- The normalizer N_G(H)={g∈ G:gHg⁻¹=H} of a subgroup Definition
- Conjugation by (1 2) in Sym({1,2,3}) exchanges the transpositions (1 3) and (2 3) Example
- A complement determines the conjugation action on the kernel Lemma
- A group extension induces a well-defined outer action on its kernel Lemma
- If K is characteristic in N and N is normal in G, then K is normal in G Lemma
- If y=g· x, then G_y=gGₓg⁻¹ Lemma
- If the kernel is complete, the extension splits over its centralizer Proposition
- Every nontrivial normal subgroup of a finite p-group meets the center nontrivially Theorem
- Recognition theorem: G=NH with N is normal in G, N∩ H=1 exactly realises an external semidirect product Theorem
- Schur-Zassenhaus existence theorem Theorem
- The conjugates of a proper subgroup do not cover a finite group Theorem
- The conjugates of H are in bijection with G/N_G(H) and, for finite G, number [G:N_G(H)] Theorem
Dependency tree · two levels
8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Milne, Group Theory, Automorphisms of Groups (standard reference, not scraped)