CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

A normal Sylow subgroup of a normal subgroup is normal in the whole group

Statement

If $N ⊴ G$ and $P$ is a normal Sylow $p$ -subgroup of $N$ , then $P ⊴ G$ . See A Sylow $p$ -subgroup is normal if and only if it is unique.

Facts & Assumptions

Given: The hypotheses and objects in the Statement.

[L1]

A Sylow $p$ -subgroup of a finite group is normal if and only if it is the unique Sylow $p$ -subgroup. (A Sylow $p$ -subgroup is normal if and only if it is unique).

[L2]

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

Proof

technique · direct

1.1

A unique Sylow subgroup is preserved by every automorphism of its ambient normal subgroup, so it is characteristic there; characteristic-in-normal gives normality in the whole group.

L1L2givenalgebra

2.1

Step 1.1 uses only that $P$ is the unique Sylow $p$ -subgroup of $N$ , so the degenerate cases are covered as well: if $p ∤ ∣ N ∣$ then $P = {1}$ , which is normal in $G$ ; if $N = {1}$ then $P = {1}$ likewise; and if $N = G$ the asserted normality is the hypothesis itself. This proves the stated claim.

step 1.1givenalgebra∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 16 results over 5 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

Keith Conrad, Consequences of the Sylow Theorems, Sections 1-5 (standard reference, not scraped)