# Verification report — KOU-21.68 (Kourovka Notebook Problem 21.68)

Verification date: 2026-09-27.

Verdict: complete counterexample to the conjecture "Semi-abelian finite groups are monomial"
(M. Kida, Kourovka Notebook, 21st issue, Problem 21.68); unrefereed.

## Statement checked
The notebook text (arXiv:1401.0300v46, updated 27 September 2026): "A finite group G is said to be semi-abelian if
it has a sequence of subgroups 1 = G_0 ⩽ G_1 ⩽ ··· ⩽ G_n = G such that for every i the subgroup G_{i+1} is
isomorphic to a quotient of a semidirect product A_i ⋊ G_i for some abelian group A_i. Conjecture: Semi-abelian
finite groups are monomial." "Monomial" is read as M-group (every irreducible complex character is induced from a
linear character of a subgroup). In v46 the problem carries no solution mark and no comment.

## Proof obligations checked
- Lemma 2.1 (Clifford correspondent of a monomial character over an abelian normal subgroup is monomial):
  derived twice independently (by the discovering agent and by the coordinating auditor) and checked line by line
  against Isaacs, Character Theory of Finite Groups (Thm 6.11, Cor 6.17, Lemma 5.11, Problem 5.3).
- Theorem 2.2 (reduction to a non-monomial ψ ∈ Irr(T)) and Corollary 2.3 (stabiliser of the coordinate
  character on the permutation module and on its augmentation submodule when [W:T] ≥ 3).
- Lemma 3.1: W = E ⋊ A4 is semi-abelian via 1 ≤ C3 ≤ A4 ≤ W; T ≅ Q8 ⋊ C3 ≅ SL(2,3) ≤ W; SL(2,3) is not an
  M-group (degree-2 irreducible characters, no subgroup of index 2).
- Every step of the sequence 1 ≤ C3 ≤ A4 ≤ W ≤ Ĝ is a genuine semidirect product with an abelian kernel, so the
  example satisfies the definition with or without the quotient clause.

## Computations (exact; scripts and outputs in reproducibility/)
1. Quaternion model built from the definitions (|W| = 96, T ≅ SL(2,3), [W:T] = 4, |Ĝ| = 768, W_{μ_T} = T).
2. Structure of the printed 16-point permutation group (normal elementary abelian B of order 8, complements,
   semidirect decompositions; stabilisers of the 7 nontrivial characters of B: four ≅ SL(2,3), three of order 32).
3. Direct non-monomiality test independent of the lemma (all subgroups of index 8 up to conjugacy; χ(1) = 8,
   ⟨χ,χ⟩ = 1, no linear constituent of χ restricted to any of them), on two realisations of the construction.
Scripts 1 and 2 were written independently of scripts 3 and 4.

## Independent adversarial audit
Verdict: CONFIRMED (2026-09-27). A separate agent was asked to break the claim. It checked the definitions
against the notebook, Dentzer's paper (via its zbMATH review) and Kida's paper, checked the proof line by line
(no gaps; one bibliographic slip in an internal note, not in the paper), and wrote an independent program
(reproducibility/5_independent_audit_full_character_table.py, output recorded). That program enumerates all
subgroups of order 96 by two methods, including a 1-cocycle classification, and computes the full character
table of Ĝ: exactly the three irreducible characters of degree 8 are non-monomial.

## Relation to the literature, novelty and scope
The conjecture is Conjecture 1.3 of M. Kida, "On semiabelian groups", J. Group Theory 28 (2025) 697–712
(open access, read in full on 2026-09-27). Kida reports that no semiabelian group of order at most 240 is
non-monomial (Magma); the example has order 768. Kida's Theorem 1.4 assumes a class closed under subgroups and
quotients consisting of semiabelian groups; it does not apply here because the subgroup T ≅ SL(2,3) is not
semiabelian (Kida himself notes a semiabelian C_2^3 ⋊ A_4 of order 96 containing SL(2,3)). With Kida's
Theorem 1.1 (isoclinism invariance) and Tappe's theorem, every group isoclinic to Ĝ is also a counterexample.
Searches on 27 September 2026 (arXiv API: semi-abelian/semiabelian with monomial/M-group, Kida, Dentzer,
Kourovka; Crossref; zbMATH) found no earlier answer, and the notebook (v46) lists the problem as open. This
negative search is not a proof of priority and no priority claim is made. The smallest order of a
semi-abelian non-M-group lies between 241 (Kida's computation) and 768 and is not determined.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/