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Unsolved Problems in Group Theory. The Kourovka Notebook
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This is a collection of open problems in group theory proposed by hundreds of mathematicians from all over the world. It has been published every 2--4 years since 1965. This is the 21st edition, which contains 150 new problems and a number of comments on problems from the previous editions.
Forward citations
Cited by 20 Pith papers
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The solution to Kourovka problem 21.88
No finite group of odd order has commuting probability 1/17; the proof uses a structural theorem making the Sylow p-subgroup normal and abelian when cp(G)=1/p.
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Products of three conjugacy classes in the alternating group
For large alternating groups, any three conjugacy classes of size at least |G|^(1-δ) multiply to the whole group for a universal small δ.
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A Torsion-free Supersoluble Group with Trivial Outer Automorphism Group
There exists a torsion-free normally poly-ℤ group of Hirsch length 14 with Out(G)=1, solving Kourovka Problem 13.23 negatively.
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The structure of finite groups invariably generated by two elements of prime order
The almost simple groups invariably generated by an element of order 2 and an element of order 3 are exactly PGL2(3^{2^b}) for b ≥ 1.
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Counterexamples to Problem 17.102 of the Kourovka Notebook: Negation and Discussion of Separability Conditions in Infinite Groups
Disjoint subsets of cardinality smaller than an infinite group need not be separable, refuting Problem 17.102.
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Hopficity of profinite completions of abelian groups
An abelian group has a topologically Hopfian profinite completion exactly when A/pA is finite for every prime p; this answers Kourovka Problem 6.30 in the negative.
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On Some Problems from the Kourovka Notebook
Eight open Kourovka Notebook problems are resolved by explicit constructions and counterexamples, with proofs claimed to be formally verified in Lean.
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Characterization of the alternating and symmetric groups by the order and conjugacy class sizes
Any finite group with the same order and conjugacy class size set as an alternating or symmetric group is isomorphic to it.
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Formal Conjectures: An Open and Evolving Benchmark for Verified Discovery in Mathematics
Formal Conjectures is a Lean 4 benchmark containing 2615 formalized problems with 1029 open conjectures, designed to evaluate automated mathematical reasoning and proof discovery.
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An improved characterisation of inner automorphisms of groups
For every group G there exists an embedding G into H such that non-trivial endomorphisms of G extend to H if and only if they are inner automorphisms.
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On a generalisation of Cameron's base size conjecture
Every 7-tuple of non-standard subgroups of a finite almost simple group has conjugates whose intersection is trivial — only M24 needs all seven.
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On the Sum of Element Orders in Finite Abelian Groups
Finite LCM-groups of equal order have equal sum of element orders precisely when they have the same order type.
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On a divisibility condition related to the sum of element orders of a finite group
A finite group G satisfies |H|-|K| divides ψ(H)-ψ(K) for all subgroups K ≤ H if and only if G is a p-group of exponent p.
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Two new functions related to the sum of element orders of a finite group
For a finite group with a normal subgroup, two new relative invariants of element orders are shown to lie on opposite sides of the corresponding invariants of the quotient, with equality cases characterized as equal o...
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A Finite Skew Brace with Perfect Additive Group and Almost Simple Multiplicative Group
A finite skew brace with additive group PSU_5(64)×A_5 and multiplicative group Aut(PSU_5(64)) is constructed, affirmatively solving Kourovka Problem 20.109.
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The Local Lifting Property, Property FD, and stability of approximate representations
3-manifold groups, limit groups, and selected one-relator and right-angled Artin groups possess the local lifting property and property FD, implying flexible stability of their approximate representations.
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Arithmetic invariants for finite simple and related groups
The paper proposes a delta-invariant encoding of element orders, class sizes, and character degrees that unifies characterization results for finite simple and related groups.
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Recognition by element orders for simple linear and unitary groups
The recognition problem by element orders is solved for every finite simple linear and unitary group.
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Kronecker classes and cliques in derangement graphs
Proves K4 exists in derangement graphs of transitive groups with degree >30, implying |G:U|≤10 for normal G in A with |A:G|=3 under union-of-conjugates condition, supporting Neumann-Praeger conjecture.
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From the Cherlin-Zilber Conjecture via sharply $2$-transitive groups to the Burnside problem
A review outlining the Cherlin-Zilber Algebraicity Conjecture, the potential role of sharply 2-transitive groups as counterexamples, and connections to the Burnside problem.
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