LemmaStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passaudited 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.

Characteristic subgroups are normal, and characteristicity is transitive

Statement

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

Facts & Assumptions

Given: Groups and subgroups satisfying the hypotheses of either assertion.

[F1]

$H char G$ means that every automorphism of $G$ maps $H$ onto itself (Characteristic subgroups).

[F2]

$H ⊴ G$ means that conjugation by every element of $G$ preserves $H$ (Normal subgroup: invariance under conjugation).

Proof

technique · direct

1.1

For each $g \in G$ , conjugation $x \mapsto g x g^{- 1}$ is an automorphism of $G$ ; if $H char G$ , [F1] says it preserves $H$ , so [F2] gives $H ⊴ G$ .

F1F2

1.2

Suppose $K char H char G$ and let $α \in Aut (G)$ . By [F1], $α (H) = H$ , so $α ∣_{H}$ is an automorphism of $H$ ; applying [F1] to $K char H$ gives $α (K) = K$ .

2.1

Since step 1.2 holds for every automorphism of $G$ , $K char G$ ; together with step 1.1 this proves both assertions.

step 1.1step 1.2F1∎

Depends on

Used by

K is normal in HcharG always implies K is normal in G False statement
The derived subgroup is characteristic and the abelianization is universal Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 7 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)