DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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.

Minimal generating sets of a group

Definition

A subset $X$ of a group is a minimal generating set when it generates and no proper subset of $X$ generates. In symbols,

$⟨ X ⟩ = G, ⟨ Y ⟩ \neq G for every Y ⊊ X,$

with generated subgroups as in The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups. Here minimal means inclusion-minimal, not minimum cardinality.

Depends on

The subgroup $\langle S \rangle$ generated by a subset, the cyclic subgroup $\langle g \rangle$, and cyclic groups

Used by

FALSE: all minimal generating sets of a finite group have the same size False statement
Burnside Basis Theorem Theorem

Dependency tree · two levels

4 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

M. van Beek, Topics in Finite p-Groups, before Theorem 3.7 (standard reference, not scraped)
K. Conrad, Generating Sets, §6 (standard reference, not scraped)