Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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 GG be a group and let S⊆GS\subseteq G. The family

NS:={N:N⊴G and S⊆N}\mathcal N_S:=\{N:N\mathrel{\trianglelefteq}G\text{ and }S\subseteq N\}

is nonempty because G⊴GG\mathrel{\trianglelefteq}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 SS in GG is

⟨ ⁣⟨S⟩ ⁣⟩G:=⋂N∈NSN.\langle\!\langle S\rangle\!\rangle_G:=\bigcap_{N\in\mathcal N_S}N.

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

Depends on

Used by

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