Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-02
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.
Inner automorphisms and
Definition
Depends on
Used by
- The outer automorphism group Out(G)=Aut(G)/Inn(G) Definition
- Conjugation by (1 2) in Sym({1,2,3}) exchanges the transpositions (1 3) and (2 3) Example
- Every inner automorphism of an abelian group is the identity Example
- An automorphism fixing the centre pointwise induces a pairing-preserving automorphism of the central quotient, with kernel the inner automorphisms Proposition
- An automorphism of an extraspecial p-group acting trivially on its Frattini quotient is inner Proposition
- Inn(G) is a normal subgroup of Aut(G) Theorem
- The map g↦(x↦ gxg⁻¹) is a homomorphism G toAut(G) with kernel Z(G) and image Inn(G) Theorem
Dependency tree · two levels
7 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)