LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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.

Distinct local full-defect blocks induce to distinct global blocks

Statement

Fix $D \leq G$ and $N = N_{G} (D)$ . If blocks $b, c$ of $k N$ have defect group $D$ and $b^{G} = c^{G}$ , then $b = c$ . No choice assumption is needed.

Facts & Assumptions

Given: The stated blocks and common induced block $B$ .

[F1]

Every local full-defect block induces to a global block of defect D proves $b^{G}$ is defined and has defect group $D$ .

[F2]

A global block of defect D determines a local block of defect D proves a global defect- $D$ block has exactly one local inducing block.

[F3]

Centralizer containment makes block induction well-defined gives the local-to-global central-character criterion.

[F4]

Modular block central characters correspond to blocks identifies a local block by its value one on its primitive idempotent.

Proof

1.1

F1 applies to $b$ and gives that the common block $B$ has defect group $D$ . Therefore F2 applies to this actual global block and subgroup, giving one primitive local idempotent $e = {Br}_{D} (f)$ , where $f$ is the idempotent of $B$ .

F1F2algebra

2.1

Both induced characters take value one at $f$ . By F3, $λ_{b} (e) = λ_{c} (e) = 1$ . Since $e$ is primitive by step 1.1, F4 forces both local blocks to be $k N e$ . Hence $b = c$ . This proves injectivity even if the set of such blocks has zero or one element. For $D = 1$ or $N = G$ it is the identity case of F2. Only finite idempotent identifications occur; no claim about the vertex of an arbitrary restriction summand is used.

F2F3F4step 1.1algebra∎

Depends on

Used by

Brauer's First Main Theorem Theorem

Dependency tree · two levels

23 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

Saunders, Modular Representation Theory, Theorem 5.16 (standard reference, not scraped)
Farrell–Lassueur, Modular Representation Theory of Finite Groups, Theorem 40.4, §40 (printed pp.8–12 of upload17) (standard reference, not scraped)