ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-04

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.

The Frattini quotient of Zp is the one-dimensional vector space Fp

Example

For the additive pro- $p$ group $Z_{p}$ , one has

$Φ (Z_{p}) = p Z_{p} and Z_{p} / Φ (Z_{p}) ≅ F_{p} .$

Facts & Assumptions

Given: The additive pro- $p$ group $Z_{p}$ .

[L1]

For a finitely generated pro- $p$ group, the Frattini subgroup is $\overline{[G, G] G^{p}}$ (For a finitely generated pro-p group, the Frattini subgroup is the closure of [G,G]G^p).

Verification

technique · direct

1.1

The additive group $Z_{p}$ is abelian, so $[Z_{p}, Z_{p}] = 0$ . Its $p$ th-power subgroup in additive notation is $p Z_{p}$ , which is already closed. Therefore [L1] gives $Φ (Z_{p}) = p Z_{p}$ .

L1givenalgebra

2.1

Reduction modulo $p$ is a surjective homomorphism $Z_{p} \to Z / p Z$ with kernel $p Z_{p}$ . By step 1.1, the quotient by $Φ (Z_{p})$ is therefore $F_{p}$ .

step 1.1algebra∎

Depends on

For a finitely generated pro-p group, the Frattini subgroup is the closure of [G,G]G^p

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Gareth Wilkes, Profinite Groups and Group Cohomology lecture notes (standard reference, not scraped)