Primitive elements of free metabelian groups

AbbreviationFree metabelian group \(M_n\) (Kourovka 14.85)#✓fully proved

For a natural number \(n\), the free metabelian group \(M_n = F_n / F_n''\) of rank \(n\): the quotient of the free group \(F_n\) on \(n\) generators by its second derived subgroup \(F_n'' = [[F_n, F_n], [F_n, F_n]]\).

abbrev FreeMetabelianGroup (n : ℕ) : Type :=
  FreeGroup (Fin n) ⧸ derivedSeries (FreeGroup (Fin n)) 2

Used by 2

Used byPrimitive elements of the free metabelian group · Primitive-preserving endomorphism problem for free metabelian groups (Kourovka 14.85)

DefinitionPrimitive elements of the free metabelian group#✓fully proved

Let \(F_n\) be the free group on \(n\) generators, let \(F_n''\) be its second derived subgroup, and let \(M_n = F_n / F_n''\) be the free metabelian group of rank \(n\). Let \(x_0, \dots, x_{n-1}\) be the images in \(M_n\) of the free generators of \(F_n\). An element \(y\) of \(M_n\) is primitive when \(y = \sigma(x_i)\) for some automorphism \(\sigma\) of \(M_n\) and some index \(i\); that is, \(y\) is the image of a standard basis element under an automorphism of \(M_n\).

def IsPrimitive {n : ℕ} (y : FreeMetabelianGroup n) : Prop :=
  ∃ (σ : FreeMetabelianGroup n ≃* FreeMetabelianGroup n) (i : Fin n),
    σ (QuotientGroup.mk (FreeGroup.of i)) = y

Uses 1 · Used by 1

Statement usesFree metabelian group \(M_n\) (Kourovka 14.85)

Used byPrimitive-preserving endomorphism problem for free metabelian groups (Kourovka 14.85)

DefinitionPrimitive-preserving endomorphism problem for free metabelian groups (Kourovka 14.85)#✓fully proved

The statement, posed by E. I. Timoshenko and V. Shpilrain, that for every natural number \(n\), every endomorphism \(\varphi\) of the free metabelian group \(M_n\) of rank \(n\) that maps every primitive element of \(M_n\) to a primitive element is bijective, hence an automorphism.

def primitivePreservingEndomorphismIsAutomorphism : Prop :=
  ∀ (n : ℕ) (φ : FreeMetabelianGroup n →* FreeMetabelianGroup n),
    (∀ y, IsPrimitive y → IsPrimitive (φ y)) → Function.Bijective φ

Uses 2 · Used by 1

Statement usesFree metabelian group \(M_n\) (Kourovka 14.85) · Primitive elements of the free metabelian group

Used byPrimitive-preserving endomorphisms of free metabelian groups are automorphisms

TheoremPrimitive-preserving endomorphisms of free metabelian groups are automorphisms#●ready to prove

For every natural number \(n\), every endomorphism \(\varphi\) of the free metabelian group \(M_n\) of rank \(n\) that maps every primitive element of \(M_n\) to a primitive element is bijective, hence an automorphism of \(M_n\). This answers Kourovka Problem 14.85 positively.

Ionin and Semidetnov (Section 1, Answer 1.2) prove that every primitive-preserving endomorphism of the free metabelian group \(M_n\) is an automorphism; the Kourovka Notebook (21st issue) answers Problem 14.85 "Yes, it is".

public theorem_wanted primitivePreservingEndomorphismIsAutomorphism_holds :
    primitivePreservingEndomorphismIsAutomorphism

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 14.85 (E. I. Timoshenko, V. Shpilrain); resolution: V. Ionin and A. Semidetnov, On some more problems from the Kourovka Notebook, preprint, arXiv:2608.29219 (2026), Section 1, Answer 1.2: every primitive-preserving endomorphism of M_n is an automorphism (the abstract states that the solutions were found with the assistance of large language models and checked by the authors); the Kourovka Notebook (21st issue) marks 14.85 solved (Yes, it is), https://arxiv.org/abs/2608.29219.

Uses 1

Statement usesPrimitive-preserving endomorphism problem for free metabelian groups (Kourovka 14.85)

3 fully proved1 ready to prove

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