False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-13

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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.

Every solvable group is abelian

Statement

False. Every solvable group is abelian.

Facts & Assumptions

Given: The symmetric group $S_{3}$ .

[F1]

A group is solvable when its derived series reaches $1$ (The derived series, solvable groups, and derived length).

[L1]

$S_{3}^{'} = A_{3}$ and $A_{3}^{'} = 1$ ( $[S_{n}, S_{n}] = A_{n}$ for $n \geq 2$ , and $[A_{n}, A_{n}] = A_{n}$ for $n \geq 5$ ).

Refutation

technique · direct

1.1

By [L1], the derived series $S_{3} > A_{3} > 1$ reaches $1$ , so $S_{3}$ is solvable by [F1].

L1F1

1.2

The transpositions $(12)$ and $(23)$ do not commute, since $(12) (23) = (123)$ while $(23) (12) = (132)$ .

algebra

2.1

Thus the solvable group $S_{3}$ is nonabelian, refuting the statement.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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)