False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 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.

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$K ⊴ H char G$ always implies $K ⊴ G$

Statement

False. If $K ⊴ H$ and $H char G$ , then $K ⊴ G$ .

Facts & Assumptions

Given: $G = A_{4}$ , its Klein four subgroup $H = V_{4}$ , and $K = ⟨ (12) (34) ⟩$ .

[F1]

A characteristic subgroup is preserved by every automorphism (Characteristic subgroups).

[L1]

Characteristic subgroups are normal, and characteristicity is transitive (Characteristic subgroups are normal, and characteristicity is transitive).

[L2]

Conjugation relabels cycle entries (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 subgroup $H$ is characteristic in $A_{4}$ : it consists of the identity together with all elements of order two, a description preserved by every automorphism.

givenF1algebra

1.2

Since $H$ is abelian, every subgroup of $H$ , including $K$ , is normal in $H$ .

givenalgebra

1.3

Conjugation by $(123) \in A_{4}$ sends $(12) (34)$ to $(23) (14)$ by [L2], and this element is not in $K$ ; hence $K ⋬ A_{4}$ .

L2algebra

2.1

Thus $K ⊴ H char G$ but $K ⋬ G$ , refuting the statement and showing why [L1] needs characteristicity at both stages.

step 1.1step 1.2step 1.3L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 8 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)

K⊴Hchar⁡GK⊴HcharG always implies K⊴GK⊴G