ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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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.
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p-sections and Brauer subsections in S3

Example

Assume the Axiom of Choice. Let $G = S_{3}$ , let $p = 2$ , and work over the residue field of a splitting $2$ -modular system. Fix a transposition $t = (12)$ . Then $S_{G} (1) = {1, (123), (132)}, S_{G} (t) = {(12), (13), (23)} .$ Moreover, $C_{G} (t) = ⟨ t ⟩ ≅ C_{2}$ has a unique block $c$ , and $c^{G} = B_{0},$ the principal block of $k S_{3}$ . Thus $(t, c)$ is a $B_{0}$ -subsection and is not a subsection for the defect-zero block $B_{1}$ . At the identity, $(1, B_{0})$ and $(1, B_{1})$ are respectively $B_{0}$ - and $B_{1}$ -subsections.

Facts & Assumptions

Given: AC, $S_{3}$ , $p = 2$ , the transposition, and the splitting system in the Example.

[F1]

A $p$ -section is determined by the conjugacy class of the unique $p$ -part (The p-section of a p-element).

[F2]

A $B$ -subsection uses local-to-global block induction (Brauer subsections and B-subsections).

[F3]

The Brauer map is coefficient projection to a centralizer, and its maximal nonzero supports are the defect groups (Brauer homomorphism for a p subgroup and Defect groups are maximal Brauer support). The principal block has Sylow defect (Principal block has sylow defect).

[F4]

For a fixed $p$ -subgroup $D$ , Brauer's First Main Theorem gives a bijection, by block induction, between the blocks of $k N_{G} (D)$ having defect group $D$ and the blocks of $k G$ having defect group $D$ (Brauer's First Main Theorem).

[F5]

A central $p$ -subgroup lies in each local defect group (Central p-subgroups lie in every block defect group). AC is available (The Axiom of Choice) and is used through the AC-stated subsection and published block contracts; the calculations below are finite.

Verification

1.1

The conjugacy classes of $S_{3}$ are the identity, the three transpositions, and the two $3$ -cycles. The identity and $3$ -cycles have $2$ -part $1$ , while the $2$ -part of a transposition is the transposition itself. All transpositions are conjugate. F1 therefore gives the two displayed sections, and these exhaust the sections indexed by conjugacy classes of $2$ -elements.

F1algebra

1.2

Direct commutation shows $C_{G} (t) = ⟨ t ⟩$ . In characteristic $2$ , $k C_{G} (t) ≅ k [X] / (X^{2} - 1) = k [X] / ((X - 1)^{2}),$ which is local: its elements $a + b (X - 1)$ are units exactly when $a \neq 0$ . It therefore has one primitive central idempotent and one block $c$ , the principal block. The group $D = ⟨ t ⟩$ is central in itself; F5 puts it in every defect group of $c$ , and since it is already the Sylow $2$ -subgroup, $D$ is the defect group of $c$ .

F5algebra

1.3

Put $a = (123)$ , $T = (12) + (13) + (23)$ , and $C = a + a^{2}$ in $k G$ . The center of $k G$ has basis $1, T, C$ , since central coefficients are constant on conjugacy classes. In characteristic $2$ one computes $T^{2} = 1 + C$ , $C^{2} = C$ , and $T C = 0$ . Hence for $z = α 1 + β T + γ C$ the equation $z^{2} = z$ forces $β = 0$ and $α, γ \in {0, 1}$ . The only central idempotents are therefore $0, 1, e : = 1 + C$ , and $f : = C$ ; thus $e, f$ are precisely the two block idempotents. The augmentation of $e$ is $1$ , so $e$ defines the principal block $B_{0}$ , while $f$ defines $B_{1}$ . F3 gives Sylow defect $D$ for $B_{0}$ . For any nontrivial $2$ -subgroup $P$ of $S_{3}$ , one has $C_{G} (P) = P$ and coefficient projection gives ${Br}_{P} (f) = 0$ , whereas ${Br}_{1} (f) = f \neq 0$ ; F3 therefore gives defect $1$ for $B_{1}$ .

F3algebra

2.1

If an element normalizes $D$ , it fixes its unique nonidentity element $t$ , and hence centralizes $t$ . Thus $N_{G} (D) = C_{G} (t) = D$ . Step 1.2 says that $c$ has defect $D$ , so F4 defines $c^{G}$ and makes it a global block of defect $D$ . Step 1.3 says that $B_{0}$ is the only such global block, because $B_{1}$ has defect $1$ . Hence $c^{G} = B_{0}$ . F2 now gives the asserted subsection statements at $t$ .

F2F4step 1.2step 1.3

3.1

For $u = 1$ , one has $C_{G} (1) = G$ , and induction from $G$ to itself fixes each block. Hence $(1, B_{i})$ is a $B_{i}$ -subsection for $i = 0, 1$ . No generalized-decomposition table is being asserted here. AC is used only through F2–F5.

F2F5algebra∎

Depends on

Used by

A p-section with no local block inducing to the chosen global block Example

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

Aschbacher–Kessar–Oliver, Fusion Systems in Algebra and Topology, Example 4.25 and Theorems 5.1 and 5.4, pp. 274–277 (standard reference, not scraped)
Craven, The Brauer Correspondence, sections 1.5–1.6, pp. 13–16 (standard reference, not scraped)