Almost regular automorphisms of p-groups

DefinitionBounded-index subgroups of bounded derived length (Kourovka 14.96)#✓fully proved

The proposition that there are functions \(f \colon \mathbb{N}^3 \to \mathbb{N}\) and \(g \colon \mathbb{N} \to \mathbb{N}\) such that for every prime \(p\), all natural numbers \(n\) and \(m\), every finite \(p\)-group \(P\) and every automorphism \(\varphi\) of \(P\) of order \(p^n\) with exactly \(p^m\) fixed points, \(P\) has a subgroup of index at most \(f(p, n, m)\) that is soluble of derived length at most \(g(m)\).

This is the question posed by E. I. Khukhro as Kourovka Problem 14.96.

def boundedIndexSubgroupOfBoundedDerivedLength : Prop :=
  ∃ (f : ℕ → ℕ → ℕ → ℕ) (g : ℕ → ℕ), ∀ p n m : ℕ, p.Prime →
    ∀ (P : Type) [Group P] [Finite P], IsPGroup p P →
      ∀ φ : MulAut P, orderOf φ = p ^ n → Nat.card {x : P // φ x = x} = p ^ m →
        ∃ H : Subgroup P, H.index ≤ f p n m ∧ derivedSeries H (g m) = ⊥

Used by 1

Used byAlmost regular automorphisms of finite \(p\)-groups and derived length (Kourovka 14.96)

TheoremAlmost regular automorphisms of finite \(p\)-groups and derived length (Kourovka 14.96)#●ready to prove

There are functions \(f \colon \mathbb{N}^3 \to \mathbb{N}\) and \(g \colon \mathbb{N} \to \mathbb{N}\) such that for every prime \(p\), all natural numbers \(n\) and \(m\), every finite \(p\)-group \(P\) and every automorphism \(\varphi\) of \(P\) of order \(p^n\) with exactly \(p^m\) fixed points, \(P\) has a subgroup of index at most \(f(p, n, m)\) that is soluble of derived length at most \(g(m)\). So Khukhro's question of Kourovka Problem 14.96 has a positive answer.

This was proved by Jaikin-Zapirain.

public theorem_wanted boundedIndexSubgroupOfBoundedDerivedLength_holds :
    boundedIndexSubgroupOfBoundedDerivedLength

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 14.96 (E. I. Khukhro); resolution: A. Jaikin-Zapirain, On almost regular automorphisms of finite p-groups, Adv. Math. 153 (2000), no. 2, 391-402, main theorem (a subgroup of (p, n, m)-bounded index and m-bounded derived length); Kourovka Archive, Problem 14.96 (answer: Yes, it is true).

Uses 1

Statement usesBounded-index subgroups of bounded derived length (Kourovka 14.96)

DefinitionBounded index and bounded class under a \(p\)-automorphism (Kourovka 10.68)#✓fully proved

The proposition that there are functions \(f(p, m)\) and \(g(m)\) of natural numbers, with natural-number values, such that for every prime \(p\), every \(m \in \mathbb{N}\), every finite \(p\)-group \(G\) and every automorphism \(\varphi\) of \(G\) of order \(p\) with exactly \(p^m\) fixed points, \(G\) has a subgroup \(H\) of index at most \(f(p, m)\) and nilpotency class at most \(g(m)\), that is, \(\gamma_{g(m)+1}(H) = 1\), where \(\gamma_1(H) = H\) and \(\gamma_{i+1}(H) = [\gamma_i(H), H]\).

def boundedIndexBoundedClassSubgroup : Prop :=
  ∃ (f : ℕ → ℕ → ℕ) (g : ℕ → ℕ), ∀ p : ℕ, p.Prime → ∀ (m : ℕ) (G : Type) [Group G] [Finite G],
    IsPGroup p G → ∀ φ : MulAut G, orderOf φ = p → Nat.card {x : G // φ x = x} = p ^ m →
      ∃ H : Subgroup G, H.index ≤ f p m ∧ H.lowerCentralSeries (g m) = ⊥

Used by 1

Used byMedvedev's theorem on automorphisms of order \(p\) with \(p^m\) fixed points

TheoremMedvedev's theorem on automorphisms of order \(p\) with \(p^m\) fixed points#●ready to prove

There are functions \(f(p, m)\) and \(g(m)\) of natural numbers, with natural-number values, such that for every prime \(p\), every \(m \in \mathbb{N}\), every finite \(p\)-group \(G\) and every automorphism \(\varphi\) of \(G\) of order \(p\) with exactly \(p^m\) fixed points, \(G\) has a subgroup \(H\) of index at most \(f(p, m)\) and nilpotency class at most \(g(m)\), that is, \(\gamma_{g(m)+1}(H) = 1\), where \(\gamma_1(H) = H\) and \(\gamma_{i+1}(H) = [\gamma_i(H), H]\).

This answers Kourovka Problem 10.68 (E. I. Khukhro) in the affirmative; the case \(m = 1\) follows from the theory of \(p\)-groups of maximal class.

public theorem_wanted boundedIndexBoundedClassSubgroup_holds : boundedIndexBoundedClassSubgroup

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 10.68 (E. I. Khukhro); resolution: Yu. Medvedev, J. London Math. Soc. (2) 59, no. 3 (1999), 787-798, Kourovka Archive, Problem 10.68 ("Yes, it is").

Uses 1

Statement usesBounded index and bounded class under a \(p\)-automorphism (Kourovka 10.68)

2 fully proved2 ready to prove

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