TheoremStatement: 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.

Every nontrivial finite $p$ -group has a normal subgroup of index $p$

Statement

Every nontrivial finite $p$ -group has a normal subgroup of index $p$ . See Every finite $p$ -group is nilpotent.

Facts & Assumptions

Given: The hypotheses and objects in the Statement.

[L1]

Every finite $p$ -group is nilpotent. The trivial group is included and has nilpotency class zero. (Every finite $p$ -group is nilpotent).

[L2]

Every maximal proper subgroup of a finite nilpotent group is normal and has prime index. (Maximal subgroups of finite nilpotent groups are normal of prime index).

[L3]

Let $G$ be a finite group and $H \leq G$ . Then $∣ G ∣ = [G : H] ∣ H ∣ .$ Consequently, under the canonical embedding $ι : N \to Z$ , $∣ H ∣$ divides $∣ G ∣$ . (Lagrange's theorem: $∣ G ∣ = [G : H] ∣ H ∣$ for every subgroup $H$ of a finite group $G$ ).

Proof

technique · direct

1.1

In the finite nonempty collection of proper subgroups choose one maximal by inclusion.

L1L2L3givenalgebra

2.1

The nilpotent maximal-subgroup theorem makes its index prime, while Lagrange makes that index divide a power of $p$ , so it is $p$ .

step 1.1givenalgebra

3.1

The maximal-subgroup theorem also makes $M$ normal, so $M$ is the required normal subgroup of index $p$ . This proves the stated claim.

step 2.1givenalgebra∎

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: 85 results over 19 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, The Sylow Theorems, Sections 1-2 (standard reference, not scraped)

Every nontrivial finite pp-group has a normal subgroup of index pp