Engel elements

DefinitionCommutator for Engel lengths#✓fully proved

For elements \(x\) and \(y\) of a group, the commutator is \([x, y] = x^{-1} y^{-1} x y\) (the convention of the Kourovka Notebook).

def kcomm {G : Type*} [Group G] (x y : G) : G := x⁻¹ * y⁻¹ * x * y

Used by 1

Used byIterated commutators for Engel lengths

DefinitionIterated commutators for Engel lengths#✓fully proved

For elements \(h\) and \(g\) of a group and a natural number \(l\), the iterated commutator \([h, g; l]\) is defined by \([h, g; 0] = h\) and \([h, g; l + 1] = [[h, g; l], g]\), where \([x, y] = x^{-1} y^{-1} x y\); in particular \([h, g; 1] = [h, g]\).

def engelComm {G : Type*} [Group G] (h g : G) : ℕ → G
  | 0 => h
  | l + 1 => kcomm (engelComm h g l) g

Uses 1 · Used by 2

Statement usesCommutator for Engel lengths

Used byEngel length · Bounded Engel element

DefinitionBounded Engel element#✓fully proved

An element \(g\) of a group \(G\) is an Engel element if there is an \(l \ge 1\) such that \([h, g; l] = 1\) for every \(h \in G\), where \([h, g; 0] = h\), \([h, g; i + 1] = [[h, g; i], g]\) and \([x, y] = x^{-1} y^{-1} x y\).

def IsEngel {G : Type*} [Group G] (g : G) : Prop :=
  ∃ l : ℕ, 1 ≤ l ∧ ∀ h : G, engelComm h g l = 1

Uses 1 · Used by 1

Statement usesIterated commutators for Engel lengths

Used byRoman'kov's problem on Engel lengths of products (Kourovka 11.88)

DefinitionEngel length#✓fully proved

For an element \(g\) of a group \(G\), the Engel length \(l(g)\) is the least \(k \ge 1\) such that \([h, g; k] = 1\) for every \(h \in G\), where \([h, g; 0] = h\), \([h, g; i + 1] = [[h, g; i], g]\) and \([x, y] = x^{-1} y^{-1} x y\). If no such \(k\) exists, that is, if \(g\) is not an Engel element, then \(l(g) = 0\).

noncomputable def engelLength {G : Type*} [Group G] (g : G) : ℕ :=
  sInf {l | 1 ≤ l ∧ ∀ h : G, engelComm h g l = 1}

Uses 1 · Used by 1

Statement usesIterated commutators for Engel lengths

Used byRoman'kov's problem on Engel lengths of products (Kourovka 11.88)

DefinitionRoman'kov's problem on Engel lengths of products (Kourovka 11.88)#✓fully proved

The statement that every group \(G\) has a polynomial \(\varphi(x, y)\) in two variables with natural-number coefficients such that \(l(uv) \le \varphi(l(u), l(v))\) for all \(u, v \in G\) for which \(u\), \(v\) and \(uv\) are Engel elements. Here \(g \in G\) is an Engel element if there is a \(k \ge 1\) with \([h, g; k] = 1\) for all \(h \in G\), and its Engel length \(l(g)\) is the least such \(k\), where \([h, g; 0] = h\), \([h, g; i + 1] = [[h, g; i], g]\) and \([x, y] = x^{-1} y^{-1} x y\).

Natural coefficients lose nothing: on \(\mathbb{N}^2\) every real polynomial is bounded by one with natural coefficients.

def engelLengthPolynomialBound : Prop :=
  ∀ (G : Type) [Group G], ∃ φ : MvPolynomial (Fin 2) ℕ, ∀ u v : G, IsEngel u → IsEngel v →
    IsEngel (u * v) → engelLength (u * v) ≤ MvPolynomial.eval ![engelLength u, engelLength v] φ

Uses 2 · Used by 1

Statement usesBounded Engel element · Engel length

Used byEngel lengths of products admit no polynomial bound (Kourovka 11.88)

TheoremEngel lengths of products admit no polynomial bound (Kourovka 11.88)#●ready to prove

There is a group \(G\) such that for every polynomial \(\varphi(x, y)\) in two variables with natural-number coefficients there are \(u, v \in G\) such that \(u\), \(v\) and \(uv\) are Engel elements and \(l(uv) > \varphi(l(u), l(v))\). Here \(g \in G\) is an Engel element if there is a \(k \ge 1\) with \([h, g; k] = 1\) for all \(h \in G\), and its Engel length \(l(g)\) is the least such \(k\), where \([h, g; 0] = h\), \([h, g; i + 1] = [[h, g; i], g]\) and \([x, y] = x^{-1} y^{-1} x y\).

For a prime \(p\), Dolbak builds finite \(p\)-groups \(G_n\) (wreath products of a cyclic group of order \(p\) by groups \(K_n\)) with elements \(c_n\) and \(c_n^{-1} b_0\) of Engel length at most \((n + 1)p - n + 1\) whose product \(b_0\) has Engel length at least \(p^n\). He takes the direct product of the \(G_n\) in the restricted sense (the direct sum, whose elements have finite support). It is a single Engel group: an element of finite support has Engel length at most the largest Engel length of its coordinates, and each \(G_n\) is a finite \(p\)-group, hence nilpotent. (The unrestricted product is not Engel: the element with \(b_0\) in every coordinate has unbounded Engel length.) Let \(u_n\) and \(v_n\) be \(c_n\) and \(c_n^{-1} b_0\) in coordinate \(n\) and \(1\) elsewhere: they are Engel elements of length at most \((n + 1)p - n + 1\), and \(u_n v_n\) has length at least \(p^n\), which exceeds every polynomial in the lengths of the factors (Theorem 1: there are Engel groups in which the Engel length of a product of two elements is bounded by no polynomial in the Engel lengths of the factors). The refutation of this statement uses only that \(u_n\), \(v_n\) and \(u_n v_n\) are Engel elements of one group. The Kourovka Archive answers 11.88 "No, it does not".

public theorem_wanted engelLengthPolynomialBound_refuted : ¬ engelLengthPolynomialBound

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 11.88 (V. A. Roman'kov); resolution: L. V. Dolbak, The Engel length of the product of Engel elements, Siberian Math. J. 47, no. 1 (2006), 55-57, Theorem 1 and its proof; Kourovka Archive, Problem 11.88: "No, it does not", https://doi.org/10.1007/s11202-006-0006-9.

Uses 1

Statement usesRoman'kov's problem on Engel lengths of products (Kourovka 11.88)

DefinitionEngel element#✓fully proved

An element \(g\) of a group \(G\) is an Engel element if for every \(h \in G\) there is an \(n \ge 0\) with \(h_n = 1\), where \(h_0 = h\) and \(h_{i + 1} = [h_i, g]\) with \([x, g] = x^{-1} g^{-1} x g\); that is, some iterated commutator \([h, g, \dots, g]\) is trivial.

def IsEngelElement {G : Type*} [Group G] (g : G) : Prop :=
  ∀ h : G, ∃ n : ℕ, (fun x => x⁻¹ * g⁻¹ * x * g)^[n] h = 1

Used by 1

Used byBludov's problem on Engel elements in simple groups (Kourovka 16.17)

DefinitionBludov's problem on Engel elements in simple groups (Kourovka 16.17)#✓fully proved

The statement that no non-abelian simple group contains an Engel element other than the identity: for every simple group \(G\) with elements \(a\), \(b\) such that \(ab \ne ba\), every Engel element \(g\) of \(G\) satisfies \(g = 1\). Here \(g\) is an Engel element if for every \(h \in G\) some iterated commutator \([h, g, \dots, g]\) is trivial, where \([x, g] = x^{-1} g^{-1} x g\).

def simpleGroupNoNontrivialEngelElement : Prop :=
  ∀ (G : Type) [Group G], IsSimpleGroup G → (∃ a b : G, a * b ≠ b * a) →
    ∀ g : G, IsEngelElement g → g = 1

Uses 1 · Used by 1

Statement usesEngel element

Used byNon-abelian simple groups can have nontrivial Engel elements (Kourovka 16.17)

TheoremNon-abelian simple groups can have nontrivial Engel elements (Kourovka 16.17)#●ready to prove

There is a non-abelian simple group \(G\) (one with elements \(a\), \(b\) such that \(ab \ne ba\)) that contains an Engel element \(g \ne 1\). Here \(g\) is an Engel element if for every \(h \in G\) some iterated commutator \([h, g, \dots, g]\) is trivial, where \([x, g] = x^{-1} g^{-1} x g\).

The Kourovka Archive answers 16.17 "No, it can" (Bludov): an involution is an Engel element in any 2-group, so non-abelian simple 2-groups are counterexamples. Simple infinite 2-groups exist by Archive 4.74(a), from the finitely generated infinite groups of exponent \(2^n\) of Ivanov and Lysënok and Zel'manov's solution of the restricted Burnside problem.

public theorem_wanted simpleGroupNoNontrivialEngelElement_refuted :
    ¬ simpleGroupNoNontrivialEngelElement

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 16.17 (V. V. Bludov); resolution: V. V. Bludov, letter of 30 March 2006, as recorded in the Kourovka Archive; infinite simple 2-groups: Kourovka Archive, Problem 4.74(a) (S. V. Ivanov, Int. J. Algebra Comput. 4 (1994), 1-308; I. G. Lysënok, Izv. Math. 60 (1996), 453-654; E. I. Zel'manov, Math. USSR-Sb. 72 (1992), 543-565), Kourovka Archive, Problem 16.17 (No, it can: an involution is an Engel element in any 2-group, and non-abelian simple 2-groups exist).

Uses 1

Statement usesBludov's problem on Engel elements in simple groups (Kourovka 16.17)

DefinitionRight Engel element#✓fully proved

For a natural number \(n\), an element \(a\) of a group \(G\) is a right \(n\)-Engel element when \([a, {}_n x] = 1\) for every \(x \in G\), where \([y, x] = y^{-1} x^{-1} y x\) and \([a, {}_n x] = [\dots[[a, x], x], \dots, x]\) is the left-normed commutator with \(n\) copies of \(x\) (so \([a, {}_0 x] = a\)).

def IsRightEngel {G : Type*} [Group G] (n : ℕ) (a : G) : Prop :=
  ∀ x : G, (fun y => y⁻¹ * x⁻¹ * y * x)^[n] a = 1

Used by 1

Used byAbdollahi's question on right Engel subgroups (Kourovka Problem 17.12)

DefinitionAbdollahi's question on right Engel subgroups (Kourovka Problem 17.12)#✓fully proved

The statement that there are functions \(e, c \colon \mathbb{N} \to \mathbb{N}\) with \(e(n) > 0\) for every \(n\) such that, for every natural number \(n\), every nilpotent group \(G\) and every normal subgroup \(H\) of \(G\) all of whose elements are right \(n\)-Engel elements of \(G\), the subgroup \(H^{e(n)}\) generated by the \(e(n)\)-th powers of elements of \(H\) is contained in the \(c(n)\)-th term \(\zeta_{c(n)}(G)\) of the upper central series of \(G\).

def rightEngelPowerUpperCentral : Prop :=
  ∃ e c : ℕ → ℕ, (∀ n, 0 < e n) ∧ ∀ (n : ℕ) (G : Type) [Group G] [Group.IsNilpotent G]
    (H : Subgroup G), H.Normal → (∀ a ∈ H, IsRightEngel n a) →
      Subgroup.closure {y | ∃ h ∈ H, y = h ^ e n} ≤ Subgroup.upperCentralSeries G (c n)

Uses 1 · Used by 1

Statement usesRight Engel element

Used byPositive answer to Abdollahi's question on right Engel subgroups (Kourovka Problem 17.12)

TheoremPositive answer to Abdollahi's question on right Engel subgroups (Kourovka Problem 17.12)#●ready to prove

There are functions \(e, c \colon \mathbb{N} \to \mathbb{N}\) with \(e(n) > 0\) for every \(n\) such that, for every natural number \(n\), every nilpotent group \(G\) and every normal subgroup \(H\) of \(G\) all of whose elements are right \(n\)-Engel elements of \(G\), the subgroup \(H^{e(n)}\) generated by the \(e(n)\)-th powers of elements of \(H\) is contained in the \(c(n)\)-th term \(\zeta_{c(n)}(G)\) of the upper central series of \(G\).

Crosby and Traustason prove that there are positive integers \(e(n)\), \(c(n)\) such that \([H^{e(n)}, {}_{c(n)} G] = 1\), that is \(H^{e(n)} \le \zeta_{c(n)}(G)\), whenever \(H\) is a normal subgroup of \(G\) consisting of right \(n\)-Engel elements and lying in some term of the upper central series; in a nilpotent group every subgroup does. The Kourovka Archive answers 17.12 "Yes, there are".

public theorem_wanted rightEngelPowerUpperCentral_holds : rightEngelPowerUpperCentral

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 17.12 (A. Abdollahi); resolution: P. G. Crosby and G. Traustason, J. Algebra 324 (2010), 875-883, main theorem as presented by G. Traustason, Right Engel subgroups (Ischia Group Theory 2010): for a normal right n-Engel subgroup H of G with H ≤ ζ_m(G) for some m, there are positive integers e(n), c(n) with [H^{e(n)}, _{c(n)} G] = 1; Kourovka Archive, Problem 17.12 (Yes, there are), http://www.dipmat2.unisa.it/ischiagrouptheory/IGT2010/talks/Traustason.pdf.

Uses 1

Statement usesAbdollahi's question on right Engel subgroups (Kourovka Problem 17.12)

9 fully proved3 ready to prove

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