False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck 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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Every subnormal series is a normal series

Statement

False. Every subnormal series is a normal series.

Facts & Assumptions

Given: In $S_{4}$ , let $V_{4} = {1, (12) (34), (13) (24), (14) (23)}$ and $K = ⟨ (12) (34) ⟩$ .

[F1]

A subnormal series requires each term to be normal only in the preceding term, while a normal series requires every term to be normal in the whole group (Subnormal and normal series, factors, refinements, and equivalence).

[L1]

Conjugating a cycle relabels each entry by the conjugating permutation (Conjugating a cycle relabels each entry: $g (a_{1} \dots a_{k}) g^{- 1} = (g (a_{1}) \dots g (a_{k}))$ ).

Refutation

technique · direct

1.1

The chain $S_{4} ▹ A_{4} ▹ V_{4} ▹ K ▹ 1$ is subnormal: $A_{4} ⊴ S_{4}$ , $V_{4} ⊴ A_{4}$ , and $K ⊴ V_{4}$ because $V_{4}$ is abelian.

givenF1algebra

1.2

Conjugation by $(123)$ sends $(12) (34)$ to $(23) (14)$ by [L1], and $(23) (14) \notin K$ ; hence $K$ is not normal in $S_{4}$ .

L1algebra

2.1

By [F1], the displayed chain is subnormal but not normal, refuting the statement.

step 1.1step 1.2F1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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)