TheoremStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passjudge 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.

Subgroups and quotients of solvable groups are solvable

Statement

Every subgroup and every quotient of a solvable group is solvable. No finiteness hypothesis is required.

Facts & Assumptions

Given: A solvable group $G$ , a subgroup $H \leq G$ , and a normal subgroup $N ⊴ G$ .

[L1]

For every $r$ , $H^{(r)} \leq G^{(r)}$ , and a surjection $f : G \to Q$ satisfies $f (G^{(r)}) = Q^{(r)}$ (Homomorphisms respect commutator subgroups and derived series).

[F1]

Solvability means $G^{(n)} = 1$ for some natural number $n$ (The derived series, solvable groups, and derived length).

[F2]

For $N ⊴ G$ , the canonical projection $q : G \to G / N$ , $q (g) = g N$ , is a surjective group homomorphism (The canonical projection $π : G \to G / N$ , $π (g) = g N$ , is a surjective group homomorphism).

Proof

technique · direct

1.1

Choose $n$ with $G^{(n)} = 1$ .

givenF1choose

2.1

By [L1], $H^{(n)} \leq G^{(n)} = 1$ , so $H$ is solvable.

step 1.1L1F1

2.2

Since the quotient map is surjective, [L1] and [F2] give $(G / N)^{(n)} = q (G^{(n)}) = 1$ , so $G / N$ is solvable.

step 1.1L1F1F2

3.1

Thus solvability passes to both subgroups and quotients.

step 2.1step 2.2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 31 results over 15 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)