The Brauer indecomposability of Scott modules with semidihedral vertex

\.Ipek Tuvay; Shigeo Koshitani

The Brauer indecomposability of Scott modules with semidihedral vertex

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that, in characteristic 2, Scott modules with semidihedral vertex are Brauer indecomposable when all fully normalized centralizers are 2-nilpotent.

desk verdict Solid, correct closing of the semidihedral case; worth a careful referee, with minor requests for self-contained centralizer facts. read the letter →

arxiv 1908.05536 v3 pith:ECOGCYTV submitted 2019-08-15 math.RT math.GR

Shigeo Koshitani , İpek Tuvay This is my paper · ORCID

classification math.RTmath.GR MSC 20C2020C0520C15

keywords BrauerindecomposabilityScottmodulessemidihedral2-groupsfusionsystemsconstruction2-nilpotentcentralizersMoritaequivalencemodularrepresentationtheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a sufficient condition for the Scott module of a finite group over a semidihedral $2$-subgroup to be Brauer indecomposable: if the fusion system $\mathcal{F}_P(G)$ is saturated and $C_G(Q)$ is $2$-nilpotent for every fully $\mathcal{F}_P(G)$-normalized nontrivial subgroup $Q$ of $P$, then the Brauer quotient $\mathrm{Sc}(G,P)(Q)$ is indecomposable as a $k[Q\,C_G(Q)]$-module for every $Q\le P$. This matters because Brauer indecomposability of $p$-permutation bimodules is a standard precondition for the gluing method that constructs splendid stable equivalences of Morita type between blocks, a route toward the abelian defect group conjecture. The result extends earlier Brauer-indecomposability theorems for abelian and dihedral vertices to the semidihedral case. A companion theorem treats the diagonal Scott module $\mathrm{Sc}(G\times G',\Delta P)$ when two groups share a semidihedral Sylow $2$-subgroup $P$ and the same fusion system, which is the bimodule form needed for such equivalences.

What carries the argument

The load-bearing mechanism is the reduction supplied by [9]: by Theorem 2.1, for saturated $\mathcal{F}_P(G)$ the Brauer indecomposability of $\mathrm{Sc}(G,P)$ is equivalent to indecomposability of the local Scott modules $\mathrm{Res}_{Q C_G(Q)}^{N_G(Q)}\,\mathrm{Sc}(N_G(Q),N_P(Q))$ for fully normalized $Q$, and Theorem 2.2 gives a sufficient condition via a subgroup $H_Q\le N_G(Q)$ for which $N_P(Q)$ is a Sylow $p$-subgroup and the index $|N_G(Q):H_Q|$ is a $p$-power. The proof feeds this reduction with a case analysis of the semidihedral group $SD_{2^n}=\langle x,y\mid x^{2^{n-1}}=y^2=1,\ y^{-1}xy=x^{2^{n-2}-1}\rangle$: the classification of its three maximal subgroups from [16], the centralizer identities $C_P(Q)=Z(Q)$ for $Q\cong C_2\times C_2$ or $Q_8$ and $C_P(Q)=Q$ for $Q=\langle xy\rangle$, the $2$-nilpotency result from [2] for $C_G(x^i)$ with $x^i\notin\{1,z\}$, and an $S_3$-subgroup construction (Lemma 2.3) used to build $H_Q$ in the exceptional cases.

What would settle it

Compute $C_P(Q)$ directly in $SD_{2^n}$ for $Q\cong C_2\times C_2$, $Q\cong Q_8$, and $Q=\langle xy\rangle$: if it is not respectively $Z(Q)$, $Z(Q)$, and $Q$, the proof's reduction fails. Alternatively, search for a finite group $G$ with semidihedral Sylow $2$-subgroup, saturated fusion system, and $2$-nilpotent $C_G(Q)$ for all fully normalized $Q$, for which $\mathrm{Sc}(G,P)$ is not Brauer indecomposable; even one such example would disprove Theorem 1.1.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: for an algebraically closed field $k$ of characteristic $2$, if $G$ has a semidihedral $2$-subgroup $P$, the fusion system $\mathcal{F}_P(G)$ is saturated, and $C_G(Q)$ is $2$-nilpotent for every fully $\mathcal{F}_P(G)$-normalized nontrivial subgroup $Q$ of $P$, then the Scott module $\mathrm{Sc}(G,P)$ is Brauer indecomposable. Theorem 1.2 extends this to $\mathrm{Sc}(G\times G',\Delta P)$, where $G$ and $G'$ have the same semidihedral Sylow $2$-subgroup $P$ and equal fusion systems $\mathcal{F}_P(G)=\mathcal{F}_P(G')$. The proof handles fully normalized $Q$ by dividing into cases: large subgroups are covered by known $2$-nilpotency of centralizers of noncentral powers in the maximal cyclic subgroup, while the exceptional subgroups $C_2\times C_2$, $C_4$, and $C_2$ are treated with explicit centralizer computations inside the semidihedral group and an $S_3$-subgroup argument.

Load-bearing premise

The theorem rests on the hypothesis that for every nontrivial fully fusion-normalized subgroup $Q$ of $P$, the centralizer $C_G(Q)$ has a normal $2$-complement; the proof also depends on the quoted identities $C_P(Q)=Z(Q)$ for $Q\cong C_2\times C_2$ or $Q_8$ and $C_P(Q)=Q$ for $Q=\langle xy\rangle$ inside semidihedral groups, and the argument collapses if those identities are wrong.

Editorial extensions

If this is right

For every group satisfying the hypotheses, each Brauer quotient $\mathrm{Sc}(G,P)(Q)$ is indecomposable as a $k[Q\,C_G(Q)]$-module, so the Scott module can act as a building block in gluing constructions for stable equivalences of Morita type.
When two groups share a semidihedral Sylow $2$-subgroup and have the same fusion system, the diagonal Scott module $\mathrm{Sc}(G\times G',\Delta P)$ is Brauer indecomposable, providing the bimodule datum needed for a splendid stable equivalence between their principal blocks.
For each fully normalized $Q$, the Brauer quotient is isomorphic to the local Scott module $\mathrm{Sc}(N_G(Q),N_P(Q))$, so global indecomposability is controlled by local indecomposability.
The result places semidihedral vertices on the same footing as abelian and dihedral vertices in the sequence of Brauer-indecomposability results for Scott modules.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof's case split suggests that for a Sylow semidihedral $2$-subgroup, Lemma 3.1 already forces $C_G(Q)$ to be $2$-nilpotent once $|Q|\ge 8$, so the actual restriction imposed by Theorem 1.1 concerns only the small subgroups $C_2$, $C_4$, and $C_2\times C_2$; proving the centralizer condition for those from fusion-theoretic axioms alone would remove the main hypothesis.
The same reduction strategy—classify subgroups, verify $2$-nilpotency except for a short list, then handle the exceptions through automorphism groups and $S_3$-subgroup constructions—looks transferable to other defect groups with known subgroup structure, such as generalized quaternion or elementary abelian $p$-groups.
A direct, self-contained proof of the quoted semidihedral centralizer identities would make Theorem 1.1 independent of structural classifications and would let the exceptional-case verification be checked mechanically for small $n$.
The diagonal result suggests a testable criterion: equality of fusion systems plus Brauer indecomposability may be enough for $\mathrm{Sc}(G\times H,\Delta P)$ to induce a stable equivalence of Morita type for semidihedral defect groups; constructing such an equivalence for an explicit pair of groups would probe the limits of the gluing method.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, this paper closes the semidihedral gap in the Brauer indecomposability program for Scott modules; previous work handled abelian and dihedral vertices, and the semidihedral case needs genuinely new case analysis around C2×C2 and Q8 subgroups. Second, the main theorems are provably correct as far as I can tell. I checked the structural facts about semidihedral 2-groups quoted from [16] directly from the presentation, and they are exactly as stated.

What's new: Theorem 1.1 gives a sufficient condition—saturated fusion system plus 2-nilpotence of centralizers of fully normalized subgroups—for Sc(G,P) to be Brauer indecomposable. Theorem 1.2 shows that if G and G′ have a common semidihedral Sylow 2-subgroup P and equal fusion systems, then the diagonal Scott module Sc(G×G′, ΔP) is Brauer indecomposable. The proof of Theorem 1.2 is the heart of the paper: Lemma 3.1 shows that any subgroup of P of order at least 8 contains an element x^i whose centralizer is 2-nilpotent by Brauer's classical theorem, and the remaining small subgroups C2, C4, and C2×C2 are handled separately. The C2 case is the most delicate, using Burry–Carlson–Puig and a Mackey argument to rule out decompositions; I found this part coherent and complete.

Soft spots, in proportion: the paper leans on unproved centralizer facts from [16]—notably that C_P(Q)=Z(Q) for Q≅C2×C2 or Q8, and C_P(Q)=Q for the order-4 subgroup . These are elementary and correct, but a referee should ask the authors to state them explicitly or include a short derivation, since the case analysis for small subgroups rests entirely on them. That is a minor presentation issue, not a substantive flaw. The paper also assumes fluency with fusion systems and Scott modules; the notation is dense but standard for the intended audience. I saw no circularity and no fitted parameters; the self-citations are to established tools rather than load-bearing prior claims.

Bottom line: this is a solid technical extension for block theory specialists. It does not reinvent the wheel, but it fills a known gap and supplies a useful tool for constructing stable equivalences of Morita type for blocks with semidihedral defect groups—relevant to Broué's conjecture. I would cite it if I worked in this area, and I would definitely send it to a specialist referee. Recommendation: accept, with minor requests for self-contained statements of the quoted centralizer facts.

Referee Report

0 major / 4 minor

Summary. The paper proves two Brauer indecomposability theorems for Scott modules with semidihedral vertex. Theorem 1.1 gives a sufficient condition in terms of the 2-nilpotence of the centralizers of all fully normalized non-trivial subgroups of a semidihedral 2-subgroup P of G, and Theorem 1.2 extends this to a diagonal Scott module Sc(G x G', Delta P) when G and G' have a common semidihedral Sylow 2-subgroup P with equal fusion systems. The proofs use the Ishioka--Kunugi reduction of Brauer indecomposability to local Scott modules, followed by a detailed case analysis of all isomorphism types of subgroups of a semidihedral 2-group.

Significance. If the results hold, they add the semidihedral case to the known families of Brauer indecomposable Scott modules (abelian and dihedral vertices), which is a relevant step for applications of the gluing method to splendid stable equivalences of Morita type. The paper is carefully structured: the main theorems are proved in full detail, the case analysis covers all fully normalized subgroups, and the arguments are traceable to published results. The centralizer facts quoted from [16] are load-bearing but are standard and are elementary consequences of the presentation of SD_{2^n}; they appear correct. I found no internal contradictions and no circular reasoning.

minor comments (4)

[Notation 1.3] The definition 'yx := yxy −1' is ambiguous; it should presumably read '{}^y x := yxy^{-1}' or use a clearer notation. Please clarify and ensure the typesetting distinguishes the two conjugation conventions.
[Proof of Theorem 1.2, Case 3] The reference 'Theorem 4.8.6 (ii)]' contains a misplaced bracket, and the phrase 'by 1.4 of [4]' is not fully standard. Please correct the citation formatting.
[Corollary 2.4 proof] The line '(K ⋊ NP (Q)) ∩ (Q CG(Q)) = (K ⋊ Q CP (Q)) = 1' is terse; adding overline notation for images in the quotient by L would greatly improve readability and make the argument easier to follow.
[Throughout] The proof relies on several centralizer facts from [16] (for example C_P(Q)=Z(Q) for Q ≅ C2×C2 or Q8, and C_P(<xy>)=<xy>). These facts are standard and correct, but since they are load-bearing a one-line verification or a remark in the text would make the paper more self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Brauer indecomposability conclusions are derived from external theorems, re-proved lemmas, and direct structural facts.

full rationale

No circular step is present. Theorem 1.1 and Theorem 1.2 are proven from independent external results: the Ishioka–Kunugi equivalence and reduction criteria (Theorems 2.1 and 2.2), Brauer's 2-nilpotence theorem for centralizers of noncentral elements (line -3 of [2, p.246]), Linckelmann's saturation and centralizer facts from [17], and elementary structural properties of semidihedral 2-groups quoted from [16]. The quoted centralizer facts, such as C_P(Q)=Z(Q) for Q isomorphic to C2 x C2 or Q8 and N_P(Q)/Q C_P(Q) ≅ C2, are directly verifiable from the presentation SD_{2^n} = <x,y | x^{2^{n-1}}=y^2=1, y^{-1}xy=x^{2^{n-2}-1}> and are not definitionally tied to the conclusion. Lemma 2.3, although originally from the authors' earlier work [13], is re-proved in the paper with a self-contained argument, so the proof does not rest on an unverified self-citation. The case analysis in Theorem 1.1 constructs the subgroup H_Q from the stated assumptions, and Theorem 1.2's Q=C2 case is handled by reducing to the already-proved cases |Q|≥4 via Mackey decomposition and Burry–Carlson–Puig, which is a legitimate induction rather than circularity. The paper does not fit parameters, rename known results as predictions, or import a uniqueness theorem from its own prior work in a load-bearing way. Self-citations such as [13], [14], [15], and [21] are contextual or methodological, not the source of the central claim. Accordingly, the derivation chain is self-contained and no circularity is found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

This paper is pure mathematics. It introduces no free parameters, no fitted values, and no invented entities. The central claims rest on a chain of previously published theorems and structural facts about semidihedral groups, listed above. The original contribution is the new case analysis and the lemmas in Sections 2 and 3.

assumptions (5)

standard math Ishioka-Kunugi reduction (Theorems 2.1 and 2.2) that Brauer indecomposability of Sc(G,P) is equivalent to indecomposability of local Scott modules over N_G(Q).
Quoted from [9]; this is the main reduction used in both proofs and is a published theorem.
standard math Structural facts about semidihedral groups from [16]: centralizers C_P(Q) as stated, three maximal subgroup types, and conjugacy classes of involutions and order-4 elements.
Used in Corollary 2.4, Lemma 3.1, and Theorem 1.2; cited without proof from [16].
standard math Brauer's result that for certain elements x^i in a group with semidihedral Sylow 2-subgroup, C_G(x^i) is 2-nilpotent.
Cited from page 246 of [2]; used in Lemma 3.1 and Theorem 1.2 for subgroups of order at least 8.
standard math Standard fusion system theory: saturation, fully normalized subgroups, and Sylow conditions in normalizers.
Used to ensure representatives exist and N_P(Q) is a Sylow subgroup of N_G(Q); cited from [1] and [17].
standard math Burry-Carlson-Puig theorem, Mackey decomposition, and p-permutation module properties.
Used in Case 3 of Theorem 1.2 to analyze vertices of direct summands; cited from [4], [18], [20].

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Pith. "Pith review of The Brauer indecomposability of Scott modules with semidihedral vertex." pith.science (2026). https://pith.science/paper/ECOGCYTV

@misc{pith2026190805536,
  author       = {Pith},
  title        = {Pith review of: The Brauer indecomposability of Scott modules with semidihedral vertex},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ECOGCYTV}},
  note         = {Machine review of arXiv:1908.05536}
}

abstract

We present a sufficient condition for the $kG$-Scott module with vertex $P$ to remain indecomposable under the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $2$, and $P$ is a semidihedral $2$-subgroup of a finite group $G$. This generalizes results for the cases where $P$ is abelian or dihedral. The Brauer indecomposability is defined \linebreak by R.~Kessar, N.~Kunugi and N.~Mitsuhashi. The motivation of \linebreak this paper is a fact that the Brauer indecomposability of a $p$-permutation bimodule ($p$ is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing method due to Brou\'e, Rickard, Linckelmann and Rouquier, that then can possibly be lifted to a splendid derived (splendid Morita) equivalence.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [16]

T. Kwon, K. Lee, I. Cho, S. Park, On ﬁnite groups with quasi-dih edral Sylow 2-groups, J. Korean Math. Soc. 17 (1980), 91–97

work page 1980

[2] [1]

Aschbacher, R

M. Aschbacher, R. Kessar, B. Oliver, Fusion Systems in Algebra and Topology , London Math. Soc. Lecture Note Series vol. 391, Cambridge University Press, Cambridge, 2011

work page 2011

[3] [2]

Brauer, Some applications of the theory of blocks of charact ers of ﬁnite groups III, J

R. Brauer, Some applications of the theory of blocks of charact ers of ﬁnite groups III, J. Algebra 3 (1966), 225–255

work page 1966

[4] [3]

Broto, R

C. Broto, R. Levi, B. Oliver, The homotopy theory of fusion syst ems, J. Amer. Math. Soc. 16 (2003), 779–856

work page 2003

[5] [4]

Brou´ e, On Scott modules andp-permutation modules: an approach through the Brauer morphism, Proc

M. Brou´ e, On Scott modules andp-permutation modules: an approach through the Brauer morphism, Proc. Amer. Math. Soc. 93 (1985), 401–408

work page 1985

[6] [5]

Brou´ e, Equivalences of blocks of group algebras, In Finite-dimensional Algebras and Related Topics (Ottawa 1992) , Eds

M. Brou´ e, Equivalences of blocks of group algebras, In Finite-dimensional Algebras and Related Topics (Ottawa 1992) , Eds. V. Dlab and L.L. Scott, Kluwer Academic Pub., Derdrecht, 1994, pp.1–26

work page 1992

[7] [6]

Brou´ e, L

M. Brou´ e, L. Puig, Characters and local structure in G-algebras, J. Algebra 63 (1980), 306–317

work page 1980

[8] [7]

Craven, A

D. Craven, A. Glesser, Fusion systems on small p-groups, Trans. Amer. Math. Soc. 364 (2012), 5945–5967

work page 2012