Alphabeta Math
DefinitionDefinition: 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.

The normal closure of a subset of a group

Definition

Let G be a group and let S⊆G. The family

NS:={N:N⊴G and S⊆N}

is nonempty because G⊴G by Normal subgroup: invariance under conjugation. Its intersection is normal by The intersection of a nonempty family of normal subgroups is normal. The normal closure of S in G is

⟨ ⁣⟨S⟩ ⁣⟩G:=⋂N∈NSN.

It contains S and is contained in every normal subgroup of G that contains S. Thus it is the smallest normal subgroup of G containing S.

Depends on

Dependency tree · 2 levels

Sources