Alphabeta MathHow 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 is nilpotent if for some , where is its upper central series (The upper central series). The least such is the nilpotency class of . This least index exists by the well-ordering principle for nonempty subsets of (The well-ordering principle).
Thus the trivial group has class . A nontrivial group has class exactly when it is abelian.
Depends on
Used by
- An extraspecial p-group is nilpotent of class exactly two and its derived subgroup has order p Corollary
- The homogeneous dimension of a finitely generated nilpotent group Definition
- A composition series and the derived series of S₃ Example
- An extension of nilpotent groups is nilpotent False statement
- Bass-Guivarch growth-degree formula Remark
- Every finite p-group is nilpotent Theorem
- Nilpotence via central series, the upper central series, and the lower central series Theorem
- Philip Hall: in a finite solvable group the Fitting subgroup contains its own centralizer Theorem
Dependency tree · two levels
13 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
- 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)