ExampleConstruction: AI-adaptedVerification: 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.

In Zp squared, topological generation is detected by the Frattini quotient coordinates

Example

In $G = Z_{p} \times Z_{p}$ , a pair $(u, v)$ topologically generates $G$ if and only if their reductions modulo $p$ form a basis of $F_{p}^{2}$ .

Facts & Assumptions

Given: The pro- $p$ group $G = Z_{p} \times Z_{p}$ .

[L1]

Topological generation in a finitely generated pro- $p$ group is detected on the Frattini quotient (A subset topologically generates a finitely generated pro-p group exactly when its image spans the Frattini quotient over Fp).

[F1]

The external direct product is coordinatewise (The external direct product $G \times H$ with componentwise multiplication).

Verification

technique · direct

1.1

By coordinatewise addition in [F1], the subgroup $p G$ consists of pairs whose two coordinates are divisible by $p$ . Therefore $G / p G ≅ (Z_{p} / p Z_{p}) \times (Z_{p} / p Z_{p}) ≅ F_{p}^{2}$ .

F1givenalgebra

2.1

The Frattini quotient of the abelian pro- $p$ group $G$ is exactly $G / p G$ , so [L1] says that a subset topologically generates $G$ exactly when its image spans $F_{p}^{2}$ . For a pair $(u, v)$ , spanning $F_{p}^{2}$ is the same as being a basis.

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Alexander Lubotzky, Combinatorial group theory for pro-p groups (standard reference, not scraped)