DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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.

Nilpotent groups and nilpotency class

Definition

A group $G$ is nilpotent if $Z_{c} (G) = G$ for some $c \in N$ , where $(Z_{r} (G))$ is its upper central series (The upper central series). The least such $c$ is the nilpotency class of $G$ . This least index exists by the well-ordering principle for nonempty subsets of $N$ (The well-ordering principle).

Thus the trivial group has class $0$ . A nontrivial group has class $1$ exactly when it is abelian.

Depends on

Used by

A composition series and the derived series of S₃ Example
An extension of nilpotent groups is nilpotent False statement
Every finite p-group is nilpotent Theorem
Nilpotence via central series, the upper central series, and the lower central series Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 33 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

J. S. Milne, Group Theory, Chapter 6 (standard reference, not scraped)
K. Conrad, Subgroup Series I (standard reference, not scraped)
K. Igusa, Notes on Jordan-Hölder, section 5 (standard reference, not scraped)