How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced — the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted — a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated — a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
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
- 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
- If y=g· x, then G_y=gGₓg⁻¹ Lemma
- Every nontrivial normal subgroup of a finite p-group meets the center nontrivially 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 · next 3 levels
Direct dependencies and their dependencies through the next three levels: 12 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Milne, Group Theory, Automorphisms of Groups (standard reference, not scraped)