LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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Central factors are equivalent to adjacent commutator containments

Statement

Let $N ⊴ G$ and $N \leq H \leq G$ . Then $H / N \leq Z (G / N) ⟺ [G, H] \leq N .$ Consequently a normal series $1 = H_{0} \leq H_{1} \leq \dots \leq H_{c} = G$ is central, meaning $H_{i + 1} / H_{i} \leq Z (G / H_{i})$ , exactly when $[G, H_{i + 1}] \leq H_{i}$ for every $0 \leq i < c$ .

Facts & Assumptions

Given: A normal subgroup $N ⊴ G$ and a subgroup $H$ containing $N$ .

[F1]

$[G, H]$ is generated by all $[g, h]$ with $g \in G$ and $h \in H$ (Subgroup commutators and the lower central series).

[F2]

An element is central exactly when it commutes with every element of the group (The center $Z (G)$ of a group).

[F3]

Multiplication in $G / N$ is $(g N) (h N) = g h N$ (The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ ).

Proof

technique · direct

1.1

Suppose $H / N \leq Z (G / N)$ . For every $g \in G$ and $h \in H$ , the cosets $g N$ and $h N$ commute by [F2]. Expanding their products with [F3] gives $g h N = h g N$ , equivalently $g h g^{- 1} h^{- 1} \in N$ ; hence [F1] gives $[G, H] \leq N$ .

assume-hypF1F2F3

2.1

Conversely, suppose $[G, H] \leq N$ . Then [F1] gives $g h g^{- 1} h^{- 1} \in N$ for all $g \in G, h \in H$ . Reversing the coset calculation in step 1.1 shows $g N$ and $h N$ commute, so $H / N \leq Z (G / N)$ by [F2].

assume-hypstep 1.1F1F2F3

3.1

Applying the equivalence of steps 1.1 and 2.1 with $(N, H) = (H_{i}, H_{i + 1})$ for each adjacent pair proves the series criterion, including $H_{0} = 1$ and $H_{c} = G$ .

step 1.1step 2.1∎

Depends on

Used by

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: 20 results over 11 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)