TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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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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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

1.1L1L2L3givenalgebra

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

2.1step 1.1givenalgebra

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

3.1step 2.1givenalgebra∎

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

Depends on

Used by

The complex numbers are algebraically closed Theorem

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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