bibkey: bahturincentronepereira2026peak
authors: Bahturin, Yuri; Centrone, Lucio; Pereira, Kauê
year: 2026
title: Quantum regularity of finite dimensional semisimple algebras
doi: 10.48550/arXiv.2608.21936
url: https://arxiv.org/abs/2608.21936v1
claim: Conjecture 31 asserts that every finite peak group is either solvable or a Frobenius group with the specified complement structure, exclusively.
strata_touched:
  - D5/S3/FiniteGroups/BahturinCentronePereiraPeakGroups
license: citation-only
triage: anchor

Peak groups and Conjecture 31

Verified locator

DOI: https://doi.org/10.48550/arXiv.2608.21936

Source: https://arxiv.org/abs/2608.21936v1

Source statements

Section 4, page 7, defines the character-degree set and peak groups:

d(G) := {deg(χ) | χ is an irreducible character of G}. We say that G is a peak group if d(G) has the greatest element d with respect to the divisibility ordering.

Definition 28, page 11, states:

Let G be a finite group with a neutral element e ∈ G and let H be a subgroup of G with {e} ⊊ H ⊊ G such that for any g ∈ G \ H, H ∩ gHg^{−1} = {e}. Then H is called a Frobenius complement in G and G is called a Frobenius group.

Conjecture 31, pages 12–13, states:

Let G be a peak group. Then, exactly one of the following holds: (1) G is a solvable group. (2) G is a Frobenius group with Frobenius complement H such that there exists a normal subgroup H0 of H/Fit(H) such that H0 ≅ SL(2, 5) × T, where T is a group satisfying the following conditions: (2.1) Every Sylow subgroup of T is cyclic. (2.2) p ∤ |T|, for any prime number p ∈ {2, 3, 5}.

The representation degrees are encoded by simple objects of FDRep ℂ G and Module.finrank ℂ. The complement condition uses H.map (MulAut.conj g) for conjugation. The Fitting subgroup is the supremum of normal nilpotent subgroups. “Exactly one” is the conjunction of the disjunction and the negation of the conjunction.

Scope

The proposed counterexample is the product of the source’s semidirect product of the additive group of the two-dimensional vector space over the field with eleven elements by SL(2, ZMod 5) with a cyclic group of order two. A nonidentity central element excludes every Frobenius complement. The representation-theoretic proof must establish peakness of this actual finite group, independently of the conjecture.

The completeness of the literature screen is ASSUMED-UNVERIFIED.