Nil-automorphisms

DefinitionThe commutator of an element with an automorphism#✓fully proved

For an automorphism \(\varphi\) of a group \(G\) and \(x \in G\), \(\delta_\varphi(x) = x^{-1} \varphi(x)\). This is the commutator \([x, \varphi]\) in the Kourovka convention \([x, y] = x^{-1} y^{-1} x y\), with \(\varphi\) acting as \(x \mapsto \varphi(x)\); the iterates \(\delta_\varphi^n(x)\) are the iterated commutators \([x, {}_n \varphi]\), with \([x, {}_0 \varphi] = x\).

def delta {G : Type*} [Group G] (φ : MulAut G) (x : G) : G := x⁻¹ * φ x

Used by 2

Used by\(e\)-automorphisms · Nil-automorphisms

DefinitionNil-automorphisms#✓fully proved

An automorphism \(\varphi\) of a group \(G\) is a nil-automorphism if for every \(a \in G\) there is \(n \ge 0\) with \(\delta_\varphi^n(a) = 1\), where \(\delta_\varphi(x) = x^{-1} \varphi(x)\); that is, some iterated commutator \([a, {}_n \varphi]\) is trivial.

def IsNilAutomorphism {G : Type*} [Group G] (φ : MulAut G) : Prop :=
  ∀ a : G, ∃ n : ℕ, (delta φ)^[n] a = 1

Uses 1 · Used by 1

Statement usesThe commutator of an element with an automorphism

Used byVilyatser's question on \(e\)-automorphisms (Kourovka 11.23)

Definition\(e\)-automorphisms#✓fully proved

An automorphism \(\varphi\) of a group \(G\) is an \(e\)-automorphism if for any two subgroups \(A\) and \(B\) of \(G\) with \(\varphi(A) = A\), \(\varphi(B) = B\) and \(A \not\subseteq B\), some \(a \in A \setminus B\) has \([a, \varphi] = a^{-1} \varphi(a) \in B\).

def IsEAutomorphism {G : Type*} [Group G] (φ : MulAut G) : Prop :=
  ∀ A B : Subgroup G, (∀ x, φ x ∈ A ↔ x ∈ A) → (∀ x, φ x ∈ B ↔ x ∈ B) → ¬ A ≤ B →
    ∃ a ∈ A, a ∉ B ∧ delta φ a ∈ B

Uses 1 · Used by 1

Statement usesThe commutator of an element with an automorphism

Used byVilyatser's question on \(e\)-automorphisms (Kourovka 11.23)

DefinitionVilyatser's question on \(e\)-automorphisms (Kourovka 11.23)#✓fully proved

The proposition that every \(e\)-automorphism \(\varphi\) of a group \(G\) is a nil-automorphism of \(G\).

def eAutomorphismIsNil : Prop :=
  ∀ (G : Type) [Group G] (φ : MulAut G), IsEAutomorphism φ → IsNilAutomorphism φ

Uses 2 · Used by 1

Statement uses\(e\)-automorphisms · Nil-automorphisms

Used byAn \(e\)-automorphism need not be a nil-automorphism

TheoremAn \(e\)-automorphism need not be a nil-automorphism#●ready to prove

Vilyatser's assertion in Kourovka Problem 11.23 is false: there are a group \(G\) and an automorphism \(\varphi\) of \(G\) that is an \(e\)-automorphism (for any subgroups \(A\) and \(B\) of \(G\) with \(\varphi(A) = A\), \(\varphi(B) = B\) and \(A \not\subseteq B\), some \(a \in A \setminus B\) has \(a^{-1} \varphi(a) \in B\)) but not a nil-automorphism: writing \(\delta(x) = x^{-1} \varphi(x)\), some \(a \in G\) has \(\delta^n(a) \ne 1\) for every \(n \ge 0\).

Muliarchyk shows that the cyclic shift of order four on the fourth direct power of the first Grigorchuk group is such an automorphism, using Bartholdi's characterization of the Engel elements of the Grigorchuk group; the preprint states that the work was carried out with assistance from ChatGPT. The Kourovka Notebook (21st issue) marks 11.23 solved, "No, not every".

public theorem_wanted eAutomorphismIsNil_refuted : ¬ eAutomorphismIsNil

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 11.23 (V. G. Vilyatser); resolution: K. Muliarchyk, An e-automorphism that is not a nil-automorphism, preprint (2026), main theorem: the cyclic shift (g₀, g₁, g₂, g₃) ↦ (g₁, g₂, g₃, g₀) on Γ⁴, Γ the first Grigorchuk group, is an e-automorphism of order four and not a nil-automorphism; the Kourovka Notebook (21st issue) marks 11.23 solved, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka_11_23_solution.pdf.

Uses 1

Statement usesVilyatser's question on \(e\)-automorphisms (Kourovka 11.23)

4 fully proved1 ready to prove

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