Camina groups

DefinitionBounded class for nilpotent Camina-type groups#✓fully proved

Kourovka Problem 17.47 (S. H. Ghate, A. S. Muktibodh) asks whether this proposition holds: there is a natural number \(c\) such that every nilpotent group \(G\) in which, for every \(x \notin [G, G]\), the coset \(x[G, G]\) of the commutator subgroup \([G, G]\) equals the conjugacy class \(x^G\) of \(x\), has nilpotency class at most \(c\).

def nilpotentCaminaClassBounded : Prop :=
  ∃ c : ℕ, ∀ (G : Type) [Group G], Group.IsNilpotent G →
    (∀ x : G, x ∉ commutator G →
      ∀ y : G, (∃ z ∈ commutator G, y = x * z) ↔ ∃ g : G, y = g * x * g⁻¹) →
    Group.nilpotencyClass G ≤ c

Used by 1

Used byNilpotent Camina-type groups have unbounded class

TheoremNilpotent Camina-type groups have unbounded class#●ready to prove

There is no natural number \(c\) such that every nilpotent group \(G\) in which, for every \(x \notin [G, G]\), the coset \(x[G, G]\) of the commutator subgroup \([G, G]\) equals the conjugacy class \(x^G\) of \(x\), has nilpotency class at most \(c\). So Kourovka Problem 17.47 has a negative answer.

Ionin and Semidetnov (Answer 4.2) construct, for every integer \(c \ge 2\), a nilpotent group of class exactly \(c\) in which \(x^G = x[G, G]\) for every \(x \notin [G, G]\). The Kourovka Notebook (21st issue) answers 17.47 "No, there is no such bound".

public theorem_wanted nilpotentCaminaClassBounded_refuted : ¬ nilpotentCaminaClassBounded

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 17.47 (S. H. Ghate, A. S. Muktibodh); resolution: V. Ionin and A. Semidetnov, On some more problems from the Kourovka Notebook, preprint, arXiv:2608.29219 (2026; solutions found with the assistance of large language models and checked by the authors, per its abstract), Answer 4.2 (for every integer c ≥ 2 there exists a nilpotent Camina group of class exactly c); the Kourovka Notebook (21st issue) marks 17.47 solved (No, there is no such bound), https://arxiv.org/abs/2608.29219.

Uses 1

Statement usesBounded class for nilpotent Camina-type groups

1 fully proved1 ready to prove

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