LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

If $K$ is characteristic in $N$ and $N$ is normal in $G$ , then $K$ is normal in $G$

Statement

If $K$ is characteristic in $N$ and $N ⊴ G$ , then $K ⊴ G$ .

Facts & Assumptions

Given: Subgroups $K \leq N \leq G$ with $K$ characteristic in $N$ and $N$ normal in $G$ .

[L1]

A characteristic subgroup is preserved by every automorphism of its ambient group (Characteristic subgroups).

[L2]

A subgroup is normal exactly when conjugation by every ambient element preserves it (Normal subgroup: invariance under conjugation).

[L3]

Conjugation by a fixed group element is an automorphism (Conjugation $x \mapsto g x g^{- 1}$ is an automorphism).

Proof

technique · direct

1.1

Fix $g \in G$ . Normality of $N$ and [L2] show that conjugation by $g$ maps $N$ to itself; by [L3], its restriction is an automorphism of $N$ .

L2L3

2.1

Since $K$ is characteristic in $N$ , [L1] gives $g K g^{- 1} = K$ . This holds for every $g \in G$ , so $K ⊴ G$ by [L2].

step 1.1L1L2∎

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: 14 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 (standard reference, not scraped)

If KK is characteristic in NN and NN is normal in GG, then KK is normal in GG