Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

For a finitely generated pro-p group, the Frattini formula retains the closure

Statement

For every finitely generated pro-p group G, one has Φ(G)=[G,G]Gp‾.

Facts & Assumptions

Given: A finitely generated pro-p group G.

[L1]

The closure-sensitive pro-p Frattini theorem states Φ(G)=[G,G]Gp‾ (For a finitely generated pro-p group, the Frattini subgroup is the closure of [G,G]G^p).

[F2]

The subgroup Gp is generated by the pth powers (The pth-power subgroup Gp).

Proof

technique · direct
1.1

The theorem [L1] applies to the finitely generated pro-p group fixed in the Statement, and the notation Gp is exactly that of [F2].

L1F2given
2.1

Therefore Φ(G)=[G,G]Gp‾ as stated. The closure cannot be discarded merely by reading the abstract-group notation [G,G]Gp.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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