ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Brauer characters of a p-group

Example

Let $P$ be a finite $p$ -group over a splitting field $k$ of characteristic $p$ . Then $P^{0} = {1}$ , every Brauer character is determined by its value at $1$ , and the unique irreducible Brauer character is the trivial character.

Facts & Assumptions

Given: A finite $p$ -group $P$ .

[F1]

Brauer characters are defined on the $p$ -regular elements (Brauer character of a finite-dimensional kG-module).

[L1]

Irreducible Brauer characters form a basis of class functions on $P^{0}$ (Irreducible Brauer characters form a basis of the p-regular class functions).

[L2]

A finite $p$ -group has only the trivial simple module in characteristic $p$ (A finite p-group has only the trivial simple module over a field of characteristic p).

Verification

technique · direct

1.1

If $g \in P$ is $p$ -regular, then $∣ g ∣$ divides the order of the $p$ -group $P$ and is also coprime to $p$ , so $∣ g ∣ = 1$ . Thus $P^{0} = {1}$ .

F1givenalgebra

2.1

By [L2], the only simple $k P$ -module is the trivial one, so the only irreducible Brauer character is the trivial character. Then [L1] is consistent with step 1.1 because the class-function space on the one-point set $P^{0}$ is one-dimensional.

L1L2step 1.1

3.1

For any finite-dimensional $k P$ -module $V$ , the Brauer character is therefore determined by $φ_{V} (1) = \dim_{k} V$ .

F1step 1.1step 2.1algebra∎

Depends on

Used by

A block with one ordinary and one Brauer character Example

Dependency tree · two levels

11 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

J. Miquel Martinez, Modular Representation Theory of Finite Groups (standard reference, not scraped)
Tudor Ciurca, Representation Theory (standard reference, not scraped)