Unsolved Problems in Group Theory. The Kourovka Notebook

The 21st edition of the Kourovka Notebook is a comprehensive collection of open problems in group theory, featuring 150 new challenges and updates on previous issues proposed by mathematicians worldwide since its inception in 1965.

The Gist

Imagine the Kourovka Notebook not as a dry academic textbook, but as a massive, ever-expanding "Wanted Poster" for the mathematical world.

Published by the Russian Academy of Sciences, this document is the 21st edition of a legendary collection of unsolved puzzles in Group Theory. To understand what that means, let's break it down using some everyday analogies.

1. What is a "Group"?

Think of a Group as a dance troupe.

• The dancers are the elements.
• The rules of how they move together (who can dance with whom, and in what order) are the operations.
• The "Group" is the entire troupe and its rulebook.

Some troupes are small and simple (like a square dance). Others are massive, chaotic, and infinite (like a mosh pit that never stops). Mathematicians study these "troupes" to understand the fundamental rules of symmetry and structure in the universe.

2. What is the "Kourovka Notebook"?

Imagine a giant, dusty bulletin board in a small village called Kourovka (near Sverdlovsk, Russia). In 1965, a mathematician pinned up the first few "Wanted" posters. These posters asked questions like:

• "Does this specific dance troupe have a secret move we haven't found yet?"
• "Can we build a troupe out of these specific rules that doesn't collapse?"

Every few years, the editors (E. I. Khukhro and V. D. Mazurov) come along, take down the old posters, and pin up new ones.

• The "Unsolved" Section: These are the current "Wanted" posters. They are problems that the smartest mathematicians in the world have tried to solve for decades and haven't cracked yet.
• The "Archive of Solved Problems": This is the "Hall of Fame." It lists the problems from the past that have been solved. It's like a museum of solved mysteries, showing the solution and the detective (mathematician) who solved it.

3. What's Inside This Specific Edition (No. 21, 2026)?

This edition is special because it's the 21st update, published in 2026. It contains 150 brand new "Wanted" posters.

Here are a few types of puzzles you might find inside, translated into plain English:

• The "Impossible Construction" Puzzle:

  • The Math: "Can we build a group with these specific rules that is both infinite and has no repeating patterns?"
  • The Analogy: Imagine trying to build a Lego tower that goes on forever but never repeats the same block pattern twice. Is it possible to do it without the tower falling apart? Some of these problems ask if such a "perfect, infinite tower" can even exist.

• The "Identity Crisis" Puzzle:

  • The Math: "If two groups look exactly the same in every way we can test them, are they actually the same group?"
  • The Analogy: Imagine two twins who look identical, have the same fingerprints, and even think the same thoughts. Are they the same person? In math, sometimes two groups act so similarly that you can't tell them apart, but deep down, they might be different. This notebook asks: "How do we know for sure?"

• The "Hidden Pattern" Puzzle:

  • The Math: "If a group has a certain property, does it have to have another property?"
  • The Analogy: If you see a bird that can fly, does it have to have feathers? In group theory, if a group behaves in a certain chaotic way, does it secretly have a hidden order? These problems try to map the connections between different types of behavior.

4. Why Does This Matter?

You might ask, "Who cares about imaginary dance troupes?"

These problems are the engine room of modern mathematics.

• Cryptography: The security of your bank account and the internet relies on the difficulty of solving certain group problems (like factoring huge numbers).
• Physics: The particles in the universe (electrons, quarks) behave according to group theory rules. Understanding these groups helps us understand the fabric of reality.
• Chemistry: The shapes of molecules are determined by symmetry groups.

5. The "Monster" and the "CFSG"

You will see a lot of references to CFSG (The Classification of Finite Simple Groups).

• The Analogy: Imagine a library that contains every possible "atomic" dance troupe (the building blocks of all other troupes). For 50 years, mathematicians worked together to catalog every single one of these atomic troupes. This catalog is the CFSG.
• The "Monster": One of the entries in this catalog is a group so huge and complex it's called "The Monster." It has more symmetries than there are atoms in the sun! The notebook often asks questions about this Monster and whether it behaves in ways we expect.

6. The "Update" Feature

One of the coolest parts of this document is that it's a living document.

• If a problem was unsolved in 2022, but someone solved it in 2024, the 2026 edition will update the entry with a "SOLVED" stamp and a link to the solution.
• It's like a video game where the "Quest Log" updates in real-time as players (mathematicians) complete the quests.

Summary

The Kourovka Notebook is the ultimate challenge list for the world's best math detectives. It collects the hardest, most mysterious questions about symmetry and structure. It doesn't just list the problems; it celebrates the history of the field by showing which ones have been solved and keeping the torch alive for the ones that remain.

It is a testament to human curiosity, showing that even after 60 years of work, there are still infinite mysteries waiting to be discovered in the world of numbers and shapes.

Technical Summary

Based on the document provided, here is a detailed technical summary of the 21st Issue of the Kourovka Notebook (2026).

1. Overview and Problem Statement

The document is the 21st Issue of the "Kourovka Notebook," a renowned collection of unsolved problems in group theory and related fields. First proposed in 1965 by M. I. Kargapolov, this collection serves as a primary communication channel for researchers in group theory.

• Core Problem: The document does not present a single research problem with a unified solution. Instead, it presents a compendium of 150 new unsolved problems (numbered 21.1 to 21.150) alongside a comprehensive archive of previously posed problems from issues 1 through 20 (1965–2022).
• Objective: To catalog the current state of knowledge in group theory, identify open questions, and stimulate research across diverse subfields including finite group theory, infinite group theory, combinatorial group theory, algebraic K-theory, and topological groups.

2. Methodology and Structure

The "methodology" of the Kourovka Notebook is that of a community-driven survey and status report.

• Compilation: The editors, E. I. Khukhro and V. D. Mazurov, solicit problems from leading mathematicians worldwide.
• Categorization: The problems are organized chronologically by the issue in which they were first proposed.
  • New Problems (2026): A dedicated section (pages 162–182) lists 150 brand-new problems.
  • Historical Archive: The bulk of the document (pages 5–161 and 183–281) lists problems from previous issues.
• Status Tracking: For each problem, the editors provide:
  • The Problem Statement: A precise mathematical formulation.
  • Context/Background: References to relevant literature, partial results, or related conjectures.
  • Current Status: Updates indicating if a problem has been solved, partially solved, or if the solution relies on the Classification of Finite Simple Groups (CFSG).
  • References: Citations to papers containing solutions or counterexamples, often including very recent preprints (2023–2026).

3. Key Contributions and Content Areas

The 21st issue covers a vast spectrum of group theory. Key thematic areas include:

A. Finite Group Theory and Character Theory

• Character Degrees and Codegrees: Problems investigate the relationship between character degrees, codegrees, and the structure of the group (e.g., solvability, simplicity).
• Prime Graphs and Spectra: Questions regarding the Gruenberg–Kegel graph (prime graph) and whether groups are uniquely determined by their element orders (spectrum).
• Block Theory: Conjectures relating defect groups of $p$-blocks to character degrees and heights (e.g., Robinson's Height Conjecture).
• Generation: Problems concerning the generation of finite simple groups by specific elements (e.g., involutions, elements of specific orders) and the "spread" of groups.

B. Infinite and Locally Finite Groups

• Burnside Problems: Investigations into free Burnside groups, periodic groups, and the Restricted Burnside Problem.
• Engel Conditions: Questions about $n$-Engel groups, whether they are nilpotent, and the structure of their radicals.
• Locally Finite Groups: Classification of simple locally finite groups, particularly those with specific centralizer conditions or Sylow subgroup structures.
• Biprimitively Finite Groups: Extensions of Shunkov's work on groups where pairs of conjugate elements of prime order generate finite subgroups.

C. Combinatorial and Geometric Group Theory

• Free Groups and Automorphisms: Problems on the automorphism groups of free groups ($Aut(F_n)$), conjugacy separability, and the decidability of equations in free groups.
• Hyperbolic Groups: Questions about the Haagerup property, acylindrical hyperbolicity, and the structure of subgroups in hyperbolic groups.
• Growth and Amenability: Investigations into groups of intermediate growth, branch groups, and the amenability of specific constructions (e.g., Thompson's group $F$).
• Braid Groups: Linearity, conjugacy separability, and automorphism groups of braid groups.

D. Topological and Profinite Groups

• Pro-$p$ Groups: Questions on cohomological dimension, finite presentations, and the structure of free pro-$p$ groups.
• Totally Disconnected Locally Compact (tdlc) Groups: Problems regarding the structure of elementary tdlc groups, decomposition ranks, and the existence of specific types of subgroups.
• Topological Groups: Issues concerning minimal topologies, compactness, and the lattice of closed subgroups.

E. Algebraic Structures and Rings

• Group Rings: Zero divisors, unit groups, and the structure of group algebras over various fields.
• Braces and Skew Braces: Connections between braces, pre-Lie algebras, and set-theoretic solutions to the Yang–Baxter equation.
• Varieties of Groups: Questions about the lattice of subvarieties, finite bases of identities, and limit varieties.

4. Key Results and Updates (2022–2026)

The 21st issue highlights significant recent breakthroughs, often marked with an asterisk (*) in the text:

• Solutions to Long-Standing Problems:
  • Problem 14.19: Confirmed that hyperbolic groups have the Noetherian Equation Property.
  • Problem 17.76: Disproved the conjecture that a finite group with exactly one non-commutator element does not exist; infinitely many such groups were found.
  • Problem 19.100: Proved that a finite group factorized by three abnormal supersoluble subgroups is supersoluble.
  • Problem 20.121: Confirmed a specific condition regarding minimal intersections of Sylow subgroups in almost simple groups of Lie type.
• Counterexamples and Negative Results:
  • Problem 18.14: The conjecture that a group with a subgroup of commutators being a subgroup for all automorphisms is nilpotent was disproved.
  • Problem 21.15: A counterexample was found showing that the amalgamated free product of symmetric groups need not embed into the symmetric group.
• Recent Preprints: The notebook heavily relies on 2024–2026 preprints (e.g., from arXiv), indicating a very active research community. Topics include new results on the spread of groups, character codegrees, and the structure of skew braces.

5. Significance

The Kourovka Notebook No. 21 is a critical resource for the mathematical community for several reasons:

1. Bibliographic Authority: It serves as the definitive index of open problems in group theory, guiding researchers toward the most pressing questions in the field.
2. Historical Record: By maintaining an archive of solved problems with references, it documents the evolution of group theory over 60 years, highlighting the impact of major theorems like the CFSG and the work of Z. Sela on free groups.
3. Interdisciplinary Bridge: The problems connect group theory with other fields such as topology (tdlc groups), logic (decidability of theories), and combinatorics (graphs and designs).
4. Catalyst for Research: The inclusion of "New Problems" for 2026 specifically targets emerging trends, such as the interaction between group theory and set-theoretic topology, and the structural properties of infinite simple groups.

In summary, this document is not a research paper with a single conclusion but a living encyclopedia of open problems that defines the frontier of group theory research as of early 2026. It reflects the depth and breadth of the field, celebrating solved milestones while rigorously outlining the challenges that remain.