Fitting height bounds from automorphisms

DefinitionKourovka Problem 19.43: Fitting height, automorphism order and centralizer size#✓fully proved

The statement that there is a function \(f\) assigning a natural number \(f(m, n)\) to every pair of natural numbers \(m\), \(n\) such that every finite soluble group \(G\) with an automorphism \(\varphi\) has Fitting height at most \(f(|\varphi|, |C_G(\varphi)|)\), where \(|\varphi|\) is the order of \(\varphi\) and \(|C_G(\varphi)|\) is the number of elements of \(G\) fixed by \(\varphi\).

The problem was posed by E. I. Khukhro.

def fittingHeightBoundedByOrderAndCentralizer : Prop :=
  ∃ f : ℕ → ℕ → ℕ, ∀ (G : Type) [Group G] [Finite G], Group.IsSolvable G → ∀ φ : MulAut G,
    FittingHeightLE G (f (orderOf φ) (Nat.card {g : G // φ g = g}))

Uses 1 · Used by 1

Statement usesGroups of Fitting height at most \(h\)

Used byFitting height is bounded by the order of an automorphism and its centralizer

TheoremFitting height is bounded by the order of an automorphism and its centralizer#●ready to prove

There is a function \(f\) assigning a natural number \(f(m, n)\) to every pair of natural numbers \(m\), \(n\) such that every finite soluble group \(G\) with an automorphism \(\varphi\) has Fitting height at most \(f(|\varphi|, |C_G(\varphi)|)\), where \(|\varphi|\) is the order of \(\varphi\) and \(|C_G(\varphi)|\) is the number of elements of \(G\) fixed by \(\varphi\).

This answers Kourovka Problem 19.43 (E. I. Khukhro) affirmatively: Khukhro proved the bound, and the 21st issue of the Kourovka Notebook marks 19.43 solved ("Yes, it is").

public theorem_wanted fittingHeightBoundedByOrderAndCentralizer_holds :
    fittingHeightBoundedByOrderAndCentralizer

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 19.43 (E. I. Khukhro); resolution: E. Khukhro, Bull. London Math. Soc. 58 (2026), no. 1, Article ID e70206, the Kourovka Notebook (21st issue) marks 19.43 solved: Yes, it is, https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesKourovka Problem 19.43: Fitting height, automorphism order and centralizer size

1 fully proved1 ready to prove

The dependency explorer needs JavaScript and its graph data.

Download the complete graph data.