Involution centralizers

DefinitionLocally finite fields (Kourovka 18.59)#✓fully proved

A field \(P\) is locally finite when, for every finite subset \(s\) of \(P\), the subfield of \(P\) generated by \(s\) is finite.

def IsLocallyFiniteField (P : Type) [Field P] : Prop :=
  ∀ s : Finset P, Finite (Subfield.closure (s : Set P))

Used by 1

Used byKourovka Problem 18.59 on involution centralizers

DefinitionKourovka Problem 18.59 on involution centralizers#✓fully proved

The proposition (Kourovka 18.59, D. V. Lytkina) that there are a periodic group \(G\) which contains an involution and in which all involutions are conjugate, and an infinite locally finite field \(P\) of characteristic \(2\), such that the centralizer of every involution \(i\) of \(G\) is isomorphic to \(\langle i \rangle \times \mathrm{PSL}_2(P)\). Here \(\langle i \rangle\) is the cyclic subgroup generated by \(i\) and \(\mathrm{PSL}_2(P) = L_2(P)\) is the quotient of \(\mathrm{SL}_2(P)\) by its centre.

The field \(P\) is chosen once for all involutions; since all involutions are conjugate, this is equivalent to allowing a field for each involution.

def periodicGroupWithPSLInvolutionCentralizers : Prop :=
  ∃ (G : Type) (_ : Group G), (∀ g : G, IsOfFinOrder g) ∧ (∃ i : G, orderOf i = 2) ∧
    (∀ i j : G, orderOf i = 2 → orderOf j = 2 → IsConj i j) ∧
    ∃ (P : Type) (_ : Field P), CharP P 2 ∧ Infinite P ∧ IsLocallyFiniteField P ∧
      ∀ i : G, orderOf i = 2 →
        Nonempty (Subgroup.centralizer {i} ≃*
          (Subgroup.zpowers i × Matrix.ProjectiveSpecialLinearGroup (Fin 2) P))

Uses 1 · Used by 1

Statement usesLocally finite fields (Kourovka 18.59)

Used byNegative answer to Kourovka Problem 18.59

TheoremNegative answer to Kourovka Problem 18.59#●ready to prove

There is no periodic group \(G\) that contains an involution, in which all involutions are conjugate, and for which there is an infinite locally finite field \(P\) of characteristic \(2\) such that the centralizer of every involution \(i\) of \(G\) is isomorphic to \(\langle i \rangle \times \mathrm{PSL}_2(P)\). Here \(\langle i \rangle\) is the cyclic subgroup generated by \(i\) and \(\mathrm{PSL}_2(P)\) is the quotient of \(\mathrm{SL}_2(P)\) by its centre.

Muliarchyk's Theorem 3 proves more: no group, periodic or not, has a single nonempty conjugacy class of involutions whose centralizers are isomorphic to \(C_2 \times \mathrm{PSL}_2(P)\) for an infinite locally finite field \(P\) of characteristic \(2\).

public theorem_wanted periodicGroupWithPSLInvolutionCentralizers_refuted :
    ¬ periodicGroupWithPSLInvolutionCentralizers

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 18.59 (D. V. Lytkina); resolution: K. Muliarchyk, Involution centralizers and a linear obstruction: Kourovka Problems 18.59 and 18.60, preprint, 2026, Theorem 3: no group has a single nonempty conjugacy class of involutions with involution centralizers isomorphic to C_2 × PSL_2(P), P an infinite locally finite field of characteristic two (periodicity not needed); the Kourovka Notebook (21st issue) marks 18.59 solved, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka_18_59-18_60_solution.pdf.

Uses 1

Statement usesKourovka Problem 18.59 on involution centralizers

DefinitionInvolutions commuting with an element#✓fully proved

For an element \(t\) of a group \(G\), this is the set of involutions of \(G\) that commute with \(t\): the elements \(s \in G\) of order \(2\) with \(st = ts\).

def commutingInvolutions {G : Type*} [Group G] (t : G) : Set G :=
  {s | orderOf s = 2 ∧ s * t = t * s}

Used by 2

Used byStrunkov's finiteness question for simple groups (Kourovka 11.96(a)) · Strunkov's question on infinite simple groups (Kourovka 11.96(b))

DefinitionStrunkov's finiteness question for simple groups (Kourovka 11.96(a))#✓fully proved

This is the statement that for every natural number \(n\) there is a bound \(N\) such that every finite simple group \(G\) containing an involution \(t\) that commutes with at most \(n\) involutions of \(G\) (counting \(t\) itself) has order at most \(N\). Bounding the order expresses "only finitely many such groups up to isomorphism".

def finitelyManySimpleFewCommutingInvolutions : Prop :=
  ∀ n : ℕ, ∃ N : ℕ, ∀ (G : Type) [Group G] [Finite G], IsSimpleGroup G →
    (∃ t : G, orderOf t = 2 ∧ (commutingInvolutions t).encard ≤ n) → Nat.card G ≤ N

Uses 1 · Used by 1

Statement usesInvolutions commuting with an element

Used byFinitely many finite simple groups with an involution commuting with few involutions

DefinitionStrunkov's question on infinite simple groups (Kourovka 11.96(b))#✓fully proved

This is the statement that for every natural number \(n\), no infinite simple group \(G\) contains an involution \(t\) that commutes with at most \(n\) involutions of \(G\) (counting \(t\) itself).

def noInfiniteSimpleFewCommutingInvolutions : Prop :=
  ∀ n : ℕ, ∀ (G : Type) [Group G], Infinite G → IsSimpleGroup G →
    ¬ ∃ t : G, orderOf t = 2 ∧ (commutingInvolutions t).encard ≤ n

Uses 1 · Used by 1

Statement usesInvolutions commuting with an element

Used byAn infinite simple group with an involution commuting with few involutions

TheoremFinitely many finite simple groups with an involution commuting with few involutions#●ready to prove

For every natural number \(n\) there is a bound \(N\) such that every finite simple group \(G\) containing an involution \(t\) that commutes with at most \(n\) involutions of \(G\) (counting \(t\) itself) has order at most \(N\).

This answers Kourovka Problem 11.96(a) in the affirmative; the Kourovka Notebook (21st issue) marks it solved, "Yes", modulo the classification of finite simple groups.

public theorem_wanted finitelyManySimpleFewCommutingInvolutions_holds :
    finitelyManySimpleFewCommutingInvolutions

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 11.96 (S. P. Strunkov); resolution: S. V. Skresanov, J. Group Theory 27, no. 6 (2024), 1197-1202, the Kourovka Notebook (21st issue) marks 11.96 solved: (a) yes (mod CFSG), and a locally finite group with such an involution is finite of n-bounded order; (b) no, PSL_2(ℝ) is an infinite simple group with an involution and no Klein four-subgroup, https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesStrunkov's finiteness question for simple groups (Kourovka 11.96(a))

TheoremAn infinite simple group with an involution commuting with few involutions#●ready to prove

There exist a natural number \(n\) and an infinite simple group \(G\) containing an involution \(t\) that commutes with at most \(n\) involutions of \(G\) (counting \(t\) itself). This answers Kourovka Problem 11.96(b) in the negative.

For example, \(\mathrm{PSL}_2(\mathbb{R})\) is an infinite simple group that contains an involution but no Klein four-subgroup, so each of its involutions commutes with no involution other than itself (\(n = 1\)); the Kourovka Notebook (21st issue) marks 11.96(b) solved, "No", citing Skresanov.

public theorem_wanted noInfiniteSimpleFewCommutingInvolutions_refuted :
    ¬ noInfiniteSimpleFewCommutingInvolutions

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 11.96 (S. P. Strunkov); resolution: S. V. Skresanov, J. Group Theory 27, no. 6 (2024), 1197-1202, the Kourovka Notebook (21st issue) marks 11.96 solved: (a) yes (mod CFSG), and a locally finite group with such an involution is finite of n-bounded order; (b) no, PSL_2(ℝ) is an infinite simple group with an involution and no Klein four-subgroup, https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesStrunkov's question on infinite simple groups (Kourovka 11.96(b))

5 fully proved3 ready to prove

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