Centralizers of maximal order

DefinitionMukhamet'yanov–Fomin problem on centralizers of maximal order#✓fully proved

For an element \(x\) of a group \(G\) write \(C_G(x)\) for its centralizer. This is the statement that for every finite group \(G\) and every element \(a \ne 1\) of \(G\) with \(|C_G(x)| \le |C_G(a)|\) for every \(x \ne 1\) in \(G\), every nilpotent subgroup \(H\) of \(G\) normalized by \(C_G(a)\) (that is, \(C_G(a)\) is contained in the normalizer of \(H\)) is contained in \(C_G(a)\).

This is Kourovka Notebook Problem 14.67 (I. T. Mukhamet'yanov, A. N. Fomin).

def maximalCentralizerContainsNormalizedNilpotent : Prop :=
  ∀ (G : Type) [Group G] [Finite G] (a : G), a ≠ 1 →
    (∀ x : G, x ≠ 1 →
      Nat.card (Subgroup.centralizer {x}) ≤ Nat.card (Subgroup.centralizer {a})) →
    ∀ H : Subgroup G, Group.IsNilpotent H →
      Subgroup.centralizer {a} ≤ Subgroup.normalizer (H : Set G) → H ≤ Subgroup.centralizer {a}

Used by 1

Used byCentralizers of maximal order need not contain normalized nilpotent subgroups

TheoremCentralizers of maximal order need not contain normalized nilpotent subgroups#●ready to prove

For an element \(x\) of a group \(G\) write \(C_G(x)\) for its centralizer. There exist a finite group \(G\), an element \(a \ne 1\) of \(G\) with \(|C_G(x)| \le |C_G(a)|\) for every \(x \ne 1\) in \(G\), and a nilpotent subgroup \(H\) of \(G\) normalized by \(C_G(a)\) (that is, \(C_G(a)\) is contained in the normalizer of \(H\)) such that \(H\) is not contained in \(C_G(a)\).

Jayadevan constructs a group \(G\) of order \(660602880\), an involution \(a\) with \(|C_G(a)| = 41287680 \ge |C_G(g)|\) for every \(g \ne 1\), and a nilpotent normal subgroup \(H\) of order \(16384\) not contained in \(C_G(a)\); the Kourovka Notebook (21st issue) answers Problem 14.67 "No, not always".

public theorem_wanted maximalCentralizerContainsNormalizedNilpotent_refuted :
    ¬ maximalCentralizerContainsNormalizedNilpotent

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 14.67 (I. T. Mukhamet'yanov, A. N. Fomin); resolution: A. Jayadevan, Centralizers of maximal order and nilpotent normal subgroups, preprint, Zenodo record 22861469 (2026), main theorem: a group of order 660602880 with an involution a and a nilpotent normal subgroup H of order 16384 such that |C_G(a)| = 41287680 ≥ |C_G(g)| for every g ≠ 1 but H is not contained in C_G(a); the Kourovka Notebook (21st issue) marks 14.67 solved (AI solution confirmed by K. Muliarchyk), https://zenodo.org/records/22861469.

Uses 1

Statement usesMukhamet'yanov–Fomin problem on centralizers of maximal order

1 fully proved1 ready to prove

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