Kourovka Notebook Problem 21.68

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A finite group $G$ is semi-abelian in the source's sense if there are subgroups $1=G_0\leq G_1\leq\cdots\leq G_m=G$ such that, for each $i<m$, some abelian group $A_i$ and action of $G_i$ on $A_i$ give a semidirect product $A_i\rtimes G_i$ admitting a surjective homomorphism onto a group isomorphic to $G_{i+1}$. A finite group is monomial if every irreducible complex character $\chi$ is induced from a one-dimensional character $\lambda$ of some subgroup $H\leq G$: $\chi=\operatorname{Ind}_H^G\lambda$. Here irreducible characters are traces of irreducible finite-dimensional complex representations and a one-dimensional character is a homomorphism $H\to\mathbb C^\times$.

Statement recorded in the supplied source.

Is every finite semi-abelian group monomial? That is, does each irreducible complex character of every group built by the specified subgroup sequence occur as induction from a linear character of a subgroup? [S2]

Short English statement

Are all finite groups constructed through the source’s abelian semidirect-product quotient process monomial?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2577 (KOU-21.68). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 M. Kida, Problem 21.68, in E. I. Khukhro and V. D. Mazurov (eds.), The Kourovka Notebook: Unsolved Problems in Group Theory, 21st edition, arXiv:1401.0300v48 (6 October 2026). See also source definitions in Problems 19.51. This version records a refutation; the displayed question is the historically registered target. https://arxiv.org/abs/1401.0300v48.