Test ranks of relatively free groups

DefinitionGroups satisfying a set of laws in countably many variables#✓fully proved

Let \(F_\mathbb{N}\) be the free group on countably many generators and \(W \subseteq F_\mathbb{N}\) a set of words, the laws. A group \(G\) satisfies the laws \(W\) when \(\varphi(w) = 1\) for every \(w \in W\) and every homomorphism \(\varphi \colon F_\mathbb{N} \to G\), that is, every law takes the value \(1\) under every substitution of elements of \(G\) for the variables. The variety defined by \(W\) is the class of these groups.

def SatisfiesLaws (W : Set (FreeGroup ℕ)) (G : Type*) [Group G] : Prop :=
  ∀ w ∈ W, ∀ φ : FreeGroup ℕ →* G, φ w = 1

Used by 2

Used byTimoshenko's test rank question, part (b) · Timoshenko's test rank question, part (a)

AbbreviationVerbal subgroup of a free group of finite rank#✓fully proved

Let \(W\) be a set of laws in the free group \(F_\mathbb{N}\) on countably many generators, and let \(F_r\) be the free group on \(r\) generators. The verbal subgroup \(V_W(F_r)\) of \(F_r\) for \(W\) is the normal closure in \(F_r\) of all elements \(\varphi(w)\) with \(w \in W\) and \(\varphi \colon F_\mathbb{N} \to F_r\) a homomorphism.

abbrev verbalSubgroup (W : Set (FreeGroup ℕ)) (r : ℕ) : Subgroup (FreeGroup (Fin r)) :=
  Subgroup.normalClosure {x | ∃ w ∈ W, ∃ φ : FreeGroup ℕ →* FreeGroup (Fin r), φ w = x}

Used by 3

Used byRelatively free group as a quotient of a free group · Timoshenko's test rank question, part (b) · Timoshenko's test rank question, part (a)

AbbreviationRelatively free group as a quotient of a free group#✓fully proved

Let \(W\) be a set of laws in the free group \(F_\mathbb{N}\) on countably many generators, and let \(r \in \mathbb{N}\). The free \(r\)-generator group \(G_r\) of the variety defined by \(W\) is the quotient \(F_r / V_W(F_r)\) of the free group \(F_r\) on \(r\) generators by its verbal subgroup \(V_W(F_r)\) for \(W\).

abbrev RelativelyFreeGroup (W : Set (FreeGroup ℕ)) (r : ℕ) : Type :=
  FreeGroup (Fin r) ⧸ verbalSubgroup W r

Uses 1 · Used by 2

Statement usesVerbal subgroup of a free group of finite rank

Used byTimoshenko's test rank question, part (b) · Timoshenko's test rank question, part (a)

DefinitionTest set of a group#✓fully proved

A subset \(S\) of a group \(G\) is a test set when every endomorphism of \(G\) that fixes every element of \(S\) is bijective, that is, an automorphism of \(G\).

def IsTestSet {G : Type*} [Group G] (S : Set G) : Prop :=
  ∀ f : G →* G, (∀ s ∈ S, f s = s) → Function.Bijective f

Used by 1

Used byTest rank of a group

DefinitionTest rank of a group#✓fully proved

A group \(G\) has test rank \(r \in \mathbb{N}\) when \(G\) has a test set of cardinality \(r\) and every test set of \(G\) has cardinality at least \(r\) (an infinite set counting as having cardinality \(\infty\)), that is, when the minimum of the cardinalities of the test sets of \(G\) is \(r\).

def HasTestRank (G : Type*) [Group G] (r : ℕ) : Prop :=
  (∃ S : Set G, IsTestSet S ∧ S.encard = r) ∧ ∀ S : Set G, IsTestSet S → (r : ℕ∞) ≤ S.encard

Uses 1 · Used by 2

Statement usesTest set of a group

Used byTimoshenko's test rank question, part (b) · Timoshenko's test rank question, part (a)

DefinitionTimoshenko's test rank question, part (a)#✓fully proved

The statement that for every set \(W\) of laws in the free group \(F_\mathbb{N}\) on countably many generators, if for every \(r \ge 1\) the free \(r\)-generator group \(G_r\) of the variety defined by \(W\) has test rank \(r\), then this variety is abelian: every group satisfying the laws \(W\) is abelian.

This is Kourovka Notebook Problem 16.87(a) (E. I. Timoshenko), which asks whether the statement holds.

def testRankVarietyAbelian : Prop :=
  ∀ W : Set (FreeGroup ℕ), (∀ r : ℕ, 1 ≤ r → HasTestRank (RelativelyFreeGroup W r) r) →
    ∀ (G : Type) [Group G], SatisfiesLaws W G → ∀ a b : G, a * b = b * a

Uses 4 · Used by 1

Statement usesTest rank of a group · Relatively free group as a quotient of a free group · Groups satisfying a set of laws in countably many variables · Verbal subgroup of a free group of finite rank

Used byNegative answer to Timoshenko's test rank question, part (a)

DefinitionTimoshenko's test rank question, part (b)#✓fully proved

The statement that for every set \(W\) of laws in the free group \(F_\mathbb{N}\) on countably many generators, if for every \(r \ge 1\) the free \(r\)-generator group \(G_r\) of the variety defined by \(W\) has test rank \(r\) and the variety is not periodic (there is no \(n > 0\) with \(g^n = 1\) for every element \(g\) of every group satisfying \(W\)), then the variety is the variety of all abelian groups: a group satisfies the laws \(W\) if and only if it is abelian.

This is Kourovka Notebook Problem 16.87(b) (E. I. Timoshenko), which asks whether the statement holds.

def testRankNonperiodicVarietyAllAbelian : Prop :=
  ∀ W : Set (FreeGroup ℕ), (∀ r : ℕ, 1 ≤ r → HasTestRank (RelativelyFreeGroup W r) r) →
    (¬ ∃ n : ℕ, 0 < n ∧ ∀ (G : Type) [Group G], SatisfiesLaws W G → ∀ g : G, g ^ n = 1) →
      ∀ (G : Type) [Group G], (SatisfiesLaws W G ↔ ∀ a b : G, a * b = b * a)

Uses 4 · Used by 1

Statement usesTest rank of a group · Relatively free group as a quotient of a free group · Groups satisfying a set of laws in countably many variables · Verbal subgroup of a free group of finite rank

Used byNon-periodic varieties with full test rank need not be abelian

TheoremNegative answer to Timoshenko's test rank question, part (a)#●ready to prove

There is a set \(W\) of laws in the free group \(F_\mathbb{N}\) on countably many generators such that for every \(r \ge 1\) the free \(r\)-generator group \(G_r\) of the variety defined by \(W\) has test rank \(r\), but the variety is not abelian: some group satisfying the laws \(W\) is not abelian.

The witness is the variety defined by \([[x, y], z] = 1\) and \([x, y]^p = 1\) for a prime \(p\), which contains the non-abelian group \(\mathrm{UT}_3(\mathbb{F}_p)\).

public theorem_wanted testRankVarietyAbelian_refuted : ¬ testRankVarietyAbelian

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 16.87 (E. I. Timoshenko); resolution: M. Kinyon and J. Urban (with Codex gpt-6-astra), Results on problems from the Kourovka Notebook, preprint, version 4 (19 September 2026), Section 3.23 (Problem 16.87): the variety M_p defined by [[x, y], z] = 1 and [x, y]^p = 1 (p prime) is nonabelian (it contains UT_3(F_p)) and nonperiodic (it contains Z), and Proposition 3.9 shows that its free group of rank r has test rank r for every r ≥ 1; the Kourovka Notebook (21st issue) marks 16.87 solved (a), b) No, it is not true in both cases; AI solution confirmed by E. I. Timoshenko), https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka-mathematics.pdf.

Uses 1

Statement usesTimoshenko's test rank question, part (a)

TheoremNon-periodic varieties with full test rank need not be abelian#●ready to prove

Resolved false: the same variety ([[x, y], z] = 1, [x, y]^p = 1, p prime) is not periodic since it contains ℤ, its free groups of every rank r ≥ 1 have test rank r (Kinyon-Urban, Proposition 3.9), and it is not the variety of all abelian groups since it contains the non-abelian group UT₃(𝔽_p). The Kourovka Notebook (21st issue) answers 16.87(b) "No" (AI solution confirmed by E. I. Timoshenko). Source: E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 16.87 (E. I. Timoshenko); resolution: M. Kinyon and J. Urban (with Codex gpt-6-astra), Results on problems from the Kourovka Notebook, preprint, version 4 (19 September 2026), Section 3.23 (Problem 16.87): the variety M_p defined by [[x, y], z] = 1 and [x, y]^p = 1 (p prime) is nonabelian (it contains UT_3(F_p)) and nonperiodic (it contains Z), and Proposition 3.9 shows that its free group of rank r has test rank r for every r ≥ 1; the Kourovka Notebook (21st issue) marks 16.87 solved (a), b) No, it is not true in both cases; AI solution confirmed by E. I. Timoshenko), https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka-mathematics.pdf.

public theorem_wanted testRankNonperiodicVarietyAllAbelian_refuted :
    ¬ testRankNonperiodicVarietyAllAbelian

Uses 1

Statement usesTimoshenko's test rank question, part (b)

7 fully proved2 ready to prove

The dependency explorer needs JavaScript and its graph data.

Download the complete graph data.