Characteristic quotients of free groups

DefinitionKourovka Problem 17.62 (two-generated characteristic quotients of free groups)#✓fully proved

The statement that for every natural number \(n \ge 3\) the free group \(F_n\) on \(n\) generators has a characteristic subgroup \(C \ne F_n\) such that the quotient \(F_n / C\) is generated by two elements, hence by fewer than \(n\) elements. This is the positive answer to Problem 17.62 (J. Conrad) in every rank where two generators are fewer than \(n\).

def characteristicQuotientFewerGenerators : Prop :=
  ∀ n : ℕ, 3 ≤ n → ∃ (C : Subgroup (FreeGroup (Fin n))) (_ : C.Characteristic), C ≠ ⊤ ∧
    ∃ a b : FreeGroup (Fin n) ⧸ C, Subgroup.closure {a, b} = ⊤

Used by 1

Used byFree groups of rank at least 3 have proper two-generated characteristic quotients

TheoremFree groups of rank at least 3 have proper two-generated characteristic quotients#●ready to prove

For every natural number \(n \ge 3\), the free group \(F_n\) on \(n\) generators has a characteristic subgroup \(C \ne F_n\) such that the quotient \(F_n / C\) is generated by two elements; since \(2 < n\), the quotient is generated by fewer than \(n\) elements.

Resolved true: Coulon and Fournier-Facio (Theorem 1.1) prove that for all \(n \ge 2\) the free group \(F_n\) admits continuum many pairwise non-isomorphic 2-generated infinite simple characteristic quotients; for \(n \ge 3\) such a quotient \(F_n / C\) is generated by fewer than \(n\) elements, and \(C\) is proper since the quotient is infinite. The Kourovka Notebook (21st issue) answers 17.62 "Yes, it is possible".

public theorem_wanted characteristicQuotientFewerGenerators_holds :
    characteristicQuotientFewerGenerators

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 17.62 (J. Conrad); resolution: R. Coulon and F. Fournier-Facio, Infinite simple characteristic quotients, preprint, arXiv:2312.11684, Theorem 1.1 (for all n ≥ 2, the free group of rank n admits continuum many pairwise non-isomorphic, 2-generated, infinite, simple, characteristic quotients; Section 1 notes that this answers Problem 17.62); the Kourovka Notebook (21st issue) marks 17.62 solved (Yes, it is possible; moreover, for all n ≥ 2, F_n admits continuum many pairwise non-isomorphic infinite simple characteristic 2-generated quotients), https://arxiv.org/abs/2312.11684.

Uses 1

Statement usesKourovka Problem 17.62 (two-generated characteristic quotients of free groups)

1 fully proved1 ready to prove

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