09-13-2026, 11:22 PM
Unsolved Problems in Group Theory: The Kourovka Notebook
E. I. Khukhro & V. D. Mazurov — 21st edition, 2026
This is not a conventional research paper but one of the most important long-running collections of open problems in group theory. The Kourovka Notebook began in 1965 after M. I. Kargapolov proposed creating a systematic collection of unsolved problems at a symposium in Kourovka, near Sverdlovsk. New editions have appeared roughly every 2–4 years, preserving earlier questions while adding new ones and recording solutions as they emerge. The current arXiv version, v46 from September 1, 2026, is the 21st edition and runs to more than 300 pages. More than 500 mathematicians have contributed problems over the Notebook's history, and more than three quarters of the questions from its first two editions have eventually been solved.
The 21st edition adds 150 new problems, while retaining unresolved questions dating all the way back to 1965. The range is remarkably broad: finite and infinite groups, simple groups, $p$-groups, free and profinite groups, group rings, automorphism groups, permutation groups, model theory, topological groups, geometric and combinatorial group theory, growth, hyperbolicity, character theory, words and verbal subgroups, and connections with structures such as braces and the Yang–Baxter equation. Some questions are extremely concrete—for example, whether certain properties of element orders characterize finite simple groups—while others concern deep structural questions such as whether particular finitely presented simple groups exist, whether automorphism groups preserve hyperbolicity properties, or how algebraic properties behave under ultraproducts. The new edition also shows how quickly the subject moves: several problems introduced in the 2026 section already have solutions or counterexamples appended to them.
An especially interesting feature of the present update is its role as a living record of mathematical problem solving rather than simply a historical catalogue. Solved questions are eventually transferred to a large Archive of Solved Problems, with references to proofs, while recently solved questions remain annotated in their original sections. The September 2026 arXiv update explicitly notes that several new solutions have been added, including some obtained using AI. For example, Problem 15.26, concerning partitions of finite $p$-groups, is now marked as solved and points to a 2026 AI-related preprint. Thus the Notebook offers an unusually clear view of how open questions pass from conjecture to solution—and, in its newest incarnation, how computational and AI-assisted mathematics is beginning to enter that process.
Key takeaways
Kourovka Notebook on arXiv
E. I. Khukhro & V. D. Mazurov — 21st edition, 2026
This is not a conventional research paper but one of the most important long-running collections of open problems in group theory. The Kourovka Notebook began in 1965 after M. I. Kargapolov proposed creating a systematic collection of unsolved problems at a symposium in Kourovka, near Sverdlovsk. New editions have appeared roughly every 2–4 years, preserving earlier questions while adding new ones and recording solutions as they emerge. The current arXiv version, v46 from September 1, 2026, is the 21st edition and runs to more than 300 pages. More than 500 mathematicians have contributed problems over the Notebook's history, and more than three quarters of the questions from its first two editions have eventually been solved.
The 21st edition adds 150 new problems, while retaining unresolved questions dating all the way back to 1965. The range is remarkably broad: finite and infinite groups, simple groups, $p$-groups, free and profinite groups, group rings, automorphism groups, permutation groups, model theory, topological groups, geometric and combinatorial group theory, growth, hyperbolicity, character theory, words and verbal subgroups, and connections with structures such as braces and the Yang–Baxter equation. Some questions are extremely concrete—for example, whether certain properties of element orders characterize finite simple groups—while others concern deep structural questions such as whether particular finitely presented simple groups exist, whether automorphism groups preserve hyperbolicity properties, or how algebraic properties behave under ultraproducts. The new edition also shows how quickly the subject moves: several problems introduced in the 2026 section already have solutions or counterexamples appended to them.
An especially interesting feature of the present update is its role as a living record of mathematical problem solving rather than simply a historical catalogue. Solved questions are eventually transferred to a large Archive of Solved Problems, with references to proofs, while recently solved questions remain annotated in their original sections. The September 2026 arXiv update explicitly notes that several new solutions have been added, including some obtained using AI. For example, Problem 15.26, concerning partitions of finite $p$-groups, is now marked as solved and points to a 2026 AI-related preprint. Thus the Notebook offers an unusually clear view of how open questions pass from conjecture to solution—and, in its newest incarnation, how computational and AI-assisted mathematics is beginning to enter that process.
Key takeaways
- The Kourovka Notebook is essentially a continuously updated database of frontier problems in group theory, spanning more than 60 years.
- Edition 21 contains 150 new problems alongside unresolved problems from previous editions and a very extensive archive of solved ones.
- It covers virtually every major branch of modern group theory, from finite simple groups and $p$-groups to geometric group theory, logic and topological groups.
- Many problems are sufficiently precise that they can become independent research projects; some survive for decades, while others are solved only months after publication.
- The 2026 update is particularly noteworthy because AI-assisted solutions have now begun to appear in the Notebook's solution history.
- For someone looking for research problems suitable for experimentation with modern AI mathematical systems, this may be one of the richest curated sources currently available.
Kourovka Notebook on arXiv
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