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 subgroup is the closure of [G,G]G^p

Statement

If $G$ is a finitely generated pro- $p$ group, then

$Φ (G) = \overline{[G, G] G^{p}},$

where $G^{p}$ is the subgroup generated by the $p$ th powers.

Facts & Assumptions

Given: A finitely generated pro- $p$ group $G$ .

[L1]

For surjective inverse limits of finite $p$ -groups, the Frattini subgroup is computed coordinatewise (For surjective inverse systems in the pro-p setting, the Frattini subgroup commutes with the inverse limit).

[L2]

For a finite $p$ -group $P$ , one has $Φ (P) = P^{'} P^{p}$ ( $Φ (P) = P^{'} P^{p}$ for a finite $p$ -group).

[F1]

The subgroup $G^{p}$ is generated by the $p$ th powers (The $p$ th-power subgroup $G^{p}$ ).

Proof

technique · direct

1.1

Let $K : = \overline{[G, G] G^{p}}$ . For every open normal subgroup $N ⊴ G$ , the finite quotient $G / N$ is a finite $p$ -group, and the image of $K$ in $G / N$ is $\overline{[G, G] G^{p}} N / N = [G / N, G / N] (G / N)^{p} = Φ (G / N)$ by [L2] and [F1].

L2F1givenalgebra

2.1

By [L1], $Φ (G)$ is the subgroup of all elements whose image in every finite quotient $G / N$ lies in $Φ (G / N)$ . Step 1.1 shows that $K$ has exactly the same image in every such quotient. Closed subgroups of a profinite group are determined by their images in all finite quotients, so $K = Φ (G)$ .

L1step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

13 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)
Alexander Lubotzky, Combinatorial group theory for pro-p groups (standard reference, not scraped)