ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Defect zero and p group boundaries

Example

Over a splitting field $k$ of characteristic $p$ , a finite $p$ -group $P$ has a single block $k P$ with defect group $P$ . The trivial group has the defect-zero block $k = M_{1} (k)$ . A nontrivial ambient defect-zero example is the block $(a + a^{2}) k S_{3} ≅ M_{2} (k)$ in characteristic $2$ , where $a = (123)$ .

Facts & Assumptions

Given: A splitting field $k$ of characteristic $p$ ; the indicated finite groups.

[F1]

A principal block has Sylow defect. (Principal block has sylow defect)

[F2]

Over a splitting field a matrix block is equivalent to a defect-zero block. (Defect zero blocks are simple algebras)

[F3]

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

Verification

technique · direct

1.1

Any nonzero finite-dimensional block algebra has a simple module: choose a proper left ideal of largest dimension; its quotient is simple since a strictly larger proper ideal would have larger dimension. Different blocks cannot have isomorphic simple $k P$ -modules, since their orthogonal central idempotents act respectively as identity and zero. By [F3] there is only one simple module, so there is only one block, necessarily the principal one. By [F1] its defect is $P$ . For $P = C_{p}$ , the explicit algebra $k [u] / ((u - 1)^{p})$ illustrates this: elements with nonzero constant coefficient in $u - 1$ are units by a finite geometric series, the rest are nilpotent, and hence its only idempotents are $0, 1$ .

F1F3

2.1

If $P = 1$ , then $k P = k$ , its sole block has defect group $1$ by step 1.1 and is $M_{1} (k)$ , in agreement with [F2]. This covers the simultaneous zero-defect and full-defect boundary without requiring a nontrivial group.

F2step 1.1

3.1

For the nontrivial example take characteristic $2$ , $a = (123)$ and $t = (12)$ . Put $f = a + a^{2}$ . Then $f^{2} = f$ , and inversion by $t$ fixes $f$ , so $f$ is central. The elements $f, f a$ are independent and span $f k ⟨ a ⟩$ , since $f a^{2} = f + f a$ ; disjoint coset supports therefore make $f, f a, f t, f a t$ a basis of $f k S_{3}$ . Define $A = (\begin{array}{cc} 0 & 1 \\ 1 & 1 \end{array})$ and $T = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ . Direct multiplication gives $A^{3} = T^{2} = I$ and $T A T = A^{- 1}$ . The presentation with these relations defines $S_{3}$ : every word reduces to $a^{i} t^{j}$ with $0 \leq i < 3$ , $0 \leq j < 2$ , and the six corresponding permutations are distinct. Thus $a \mapsto A, t \mapsto T$ extends to an algebra map $ρ : k S_{3} \to M_{2} (k)$ . It sends $f$ to $A + A^{2} = I$ , and the displayed basis of $f k S_{3}$ to $I, A, T, A T$ . These span all matrices, since $E_{22} = A + T$ , $E_{11} = I + A + T$ , $E_{21} = A T + I$ , and $E_{12} = T + A T + I$ . Hence the restricted map is a surjection between four-dimensional algebras, and is an isomorphism. Its scalar centre makes $f$ primitive central, and [F2] gives defect $1$ . This repeats the calculation within this example over every characteristic-two field, without a root-of-unity assumption or a dependency on another example.

F2∎

Sources

Webb, A Course in Finite Group Representation Theory, §§11.3, 11.6 and 12.3–12.5, especially pp.240–245. Local argument and conventions as displayed above.

Depends on

Used by

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Sources

Webb, A Course in Finite Group Representation Theory, §§11.3, 11.6 and 12.3–12.5, especially pp.240–245 (standard reference, not scraped)