TheoremStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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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.
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A central extension of a class- $c$ nilpotent group is nilpotent of class at most $c + 1$

Statement

Let $N ⊴ G$ with $N \leq Z (G)$ . If $G / N$ is nilpotent of class at most $c$ , then $G$ is nilpotent of class at most $c + 1$ .

Facts & Assumptions

Given: A central normal subgroup $N ⊴ G$ and an integer $c \geq 0$ such that $G / N$ has class at most $c$ .

[F1]

$γ_{1} (H) = H$ and $γ_{r + 1} (H) = [H, γ_{r} (H)]$ (Subgroup commutators and the lower central series).

[L1]

For every group $H$ and natural number $d$ , the conditions that $H$ has a central series of length $d$ , that $Z_{d} (H) = H$ , and that $γ_{d + 1} (H) = 1$ are equivalent; the least such $d$ is the nilpotency class (Nilpotence via central series, the upper central series, and the lower central series).

Proof

technique · direct

1.1

For the quotient map $q : G \to G / N$ , induction from [F1] gives $q (γ_{r} (G)) = γ_{r} (G / N)$ for every $r$ , because $q$ is surjective and sends commutators onto commutators.

F1algebra

2.1

If $G / N$ has class $e \leq c$ , then [L1] gives $γ_{e + 1} (G / N) = 1$ , and [F1] keeps all later lower-central terms trivial; hence $γ_{c + 1} (G / N) = 1$ . Step 1.1 therefore gives $γ_{c + 1} (G) \leq N$ .

givenstep 1.1F1L1

3.1

Centrality of $N$ gives $[G, N] = 1$ , so $γ_{c + 2} (G) = [G, γ_{c + 1} (G)] \leq [G, N] = 1$ .

step 2.1F1algebra

4.1

By [L1], $G$ is nilpotent of class at most $c + 1$ . The case $c = 0$ is included: then $G / N = 1$ , so $G = N \leq Z (G)$ and $G$ has class at most one.

step 3.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)

A central extension of a class-cc nilpotent group is nilpotent of class at most c+1c+1