Prime-order automorphisms with centralizer of finite rank

DefinitionGroups of bounded rank#✓fully proved

A group \(G\) has rank at most \(r\), for a natural number \(r\), when every finitely generated subgroup of \(G\) is generated by some set of at most \(r\) elements.

For periodic nilpotent groups this rank is the sectional rank.

def RankLE (G : Type*) [Group G] (r : ℕ) : Prop :=
  ∀ H : Subgroup G, H.FG → ∃ S : Finset G, S.card ≤ r ∧ Subgroup.closure (S : Set G) = H

Used by 1

Used byNilpotent-by-bounded-rank problem for automorphisms of prime order (Kourovka 13.58)

DefinitionFixed-point subgroup of an automorphism#✓fully proved

For an automorphism \(\varphi\) of a group \(G\), the centralizer \(C_G(\varphi)\) is the subgroup \(\{x \in G : \varphi(x) = x\}\) of fixed points of \(\varphi\).

def fixedSubgroup {G : Type*} [Group G] (φ : G ≃* G) : Subgroup G :=
  (φ : G →* G).eqLocus (MonoidHom.id G)

Used by 1

Used byNilpotent-by-bounded-rank problem for automorphisms of prime order (Kourovka 13.58)

DefinitionNilpotent-by-bounded-rank problem for automorphisms of prime order (Kourovka 13.58)#✓fully proved

The statement that there are functions \(f \colon \mathbb{N} \to \mathbb{N}\) and \(g \colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}\) with the following property. Let \(G\) be a nilpotent group in which every element has finite order, let \(\varphi\) be an automorphism of \(G\) of prime order \(p\), and let \(r\) be a natural number such that the fixed-point subgroup \(C_G(\varphi)\) has rank at most \(r\), that is, every finitely generated subgroup of \(C_G(\varphi)\) is generated by at most \(r\) elements. Then \(G\) has a normal subgroup \(N\), nilpotent of class at most \(f(p)\), such that the quotient \(G/N\) has rank at most \(g(r, p)\).

For periodic nilpotent groups this rank is the sectional rank.

def nilpotentByBoundedRankOfCentralizerRank : Prop :=
  ∃ (f : ℕ → ℕ) (g : ℕ → ℕ → ℕ), ∀ (G : Type) [Group G], Group.IsNilpotent G →
    (∀ x : G, IsOfFinOrder x) → ∀ (φ : G ≃* G) (p r : ℕ), p.Prime → orderOf φ = p →
      RankLE (fixedSubgroup φ) r →
      ∃ (N : Subgroup G) (_ : N.Normal), Subgroup.upperCentralSeries N (f p) = ⊤ ∧
        RankLE (G ⧸ N) (g r p)

Uses 2 · Used by 1

Statement usesGroups of bounded rank · Fixed-point subgroup of an automorphism

Used byKhukhro's theorem on automorphisms of prime order with centralizer of finite rank

TheoremKhukhro's theorem on automorphisms of prime order with centralizer of finite rank#●ready to prove

There are functions \(f \colon \mathbb{N} \to \mathbb{N}\) and \(g \colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}\) with the following property. Let \(G\) be a nilpotent group in which every element has finite order, let \(\varphi\) be an automorphism of \(G\) of prime order \(p\), and let \(r\) be a natural number such that the fixed-point subgroup \(C_G(\varphi)\) has rank at most \(r\), that is, every finitely generated subgroup of \(C_G(\varphi)\) is generated by at most \(r\) elements. Then \(G\) has a normal subgroup \(N\), nilpotent of class at most \(f(p)\), such that the quotient \(G/N\) has rank at most \(g(r, p)\). This is the positive answer to Kourovka Problem 13.58; the case \(p = 2\) was proved earlier by Shumyatsky.

public theorem_wanted nilpotentByBoundedRankOfCentralizerRank_holds :
    nilpotentByBoundedRankOfCentralizerRank

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 13.58 (E. I. Khukhro); resolution: E. I. Khukhro, J. London Math. Soc. 77 (2008), 130-148 (the case p = 2: P. Shumyatsky, Arch. Math. 71 (1998), 425-432), Kourovka Archive, Problem 13.58 (answer: Yes, it does).

Uses 1

Statement usesNilpotent-by-bounded-rank problem for automorphisms of prime order (Kourovka 13.58)

3 fully proved1 ready to prove

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