Transitive automorphism groups of elementary abelian groups

DefinitionLocally finite field (Kourovka 18.60)#✓fully proved

A field \(P\) is locally finite when the subfield generated by any finite subset of \(P\) is finite.

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

Used by 1

Used byKourovka Problem 18.60 (field-like transitive automorphism groups)

DefinitionKourovka Problem 18.60 (field-like transitive automorphism groups)#✓fully proved

The conjecture of D. V. Lytkina, as a proposition: for every countably infinite elementary abelian \(2\)-group \(V\) (an abelian group with \(v + v = 0\) for all \(v \in V\)) there is no subgroup \(G\) of the symmetric group \(\mathrm{Sym}(V)\) consisting of additive bijections such that (a) \(G\) is transitive on \(V \setminus \{0\}\), and (b) for every nonzero \(v \in V\), writing \(H = G_v\) for the stabilizer of \(v\) in \(G\), there are a subgroup \(V_v\) of \(V\) with \(V = \langle v \rangle \oplus V_v\) that every element of \(H\) maps into itself, a locally finite field \(P\) of characteristic \(2\), a group isomorphism \(\psi : H \to P^\times\) onto the multiplicative group of \(P\), and a bijection \(\theta : V_v \to P\) such that \(\theta(h w) = \psi(h)\,\theta(w)\) for all \(h \in H\) and \(w \in V_v\).

def noTransitiveFieldLikeAutomorphismGroup : Prop :=
  ∀ (V : Type) [AddCommGroup V], Countable V → Infinite V → (∀ v : V, v + v = 0) →
    ¬ ∃ G : Subgroup (Equiv.Perm V), (∀ g ∈ G, ∀ a b : V, g (a + b) = g a + g b) ∧
      (∀ u w : V, u ≠ 0 → w ≠ 0 → ∃ g ∈ G, g u = w) ∧
      ∀ v : V, v ≠ 0 → ∃ (W : AddSubgroup V) (P : Type) (_ : Field P),
        CharP P 2 ∧ IsLocallyFiniteField P ∧ IsCompl (AddSubgroup.zmultiples v) W ∧
        (∀ g ∈ G, g v = v → ∀ w ∈ W, g w ∈ W) ∧
        ∃ (ψ : ↥(G ⊓ MulAction.stabilizer (Equiv.Perm V) v) ≃* Pˣ) (θ : V → P),
          Set.BijOn θ (W : Set V) Set.univ ∧
          ∀ h : ↥(G ⊓ MulAction.stabilizer (Equiv.Perm V) v), ∀ w ∈ W,
            θ ((h : Equiv.Perm V) w) = (ψ h : P) * θ w

Uses 1 · Used by 1

Statement usesLocally finite field (Kourovka 18.60)

Used byNo field-like transitive automorphism group of a countably infinite elementary abelian \(2\)-group

TheoremNo field-like transitive automorphism group of a countably infinite elementary abelian \(2\)-group#●ready to prove

No field-like transitive automorphism group of a countably infinite elementary abelian \(2\)-group. The conjecture of Kourovka Problem 18.60 holds: for every countably infinite elementary abelian \(2\)-group \(V\) (an abelian group with \(v + v = 0\) for all \(v \in V\)) there is no subgroup \(G\) of the symmetric group \(\mathrm{Sym}(V)\) consisting of additive bijections such that (a) \(G\) is transitive on \(V \setminus \{0\}\), and (b) for every nonzero \(v \in V\), writing \(H = G_v\) for the stabilizer of \(v\) in \(G\), there are a subgroup \(V_v\) of \(V\) with \(V = \langle v \rangle \oplus V_v\) that every element of \(H\) maps into itself, a locally finite field \(P\) of characteristic \(2\), a group isomorphism \(\psi : H \to P^\times\) onto the multiplicative group of \(P\), and a bijection \(\theta : V_v \to P\) such that \(\theta(h w) = \psi(h)\,\theta(w)\) for all \(h \in H\) and \(w \in V_v\).

Muliarchyk's Theorem 1 excludes a group transitive on the nonzero vectors of an \(\mathbb{F}_2\)-space of dimension at least \(4\) whose point stabilizer preserves a complement to \(\mathbb{F}_2 v\) and is transitive on its nonzero vectors; a similarity with \(P^\times\) acting on \(P\) gives that transitivity (Corollary 2).

public theorem_wanted noTransitiveFieldLikeAutomorphismGroup_holds :
    noTransitiveFieldLikeAutomorphismGroup

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 18.60 (D. V. Lytkina); resolution: K. Muliarchyk, Involution centralizers and a linear obstruction: Kourovka Problems 18.59 and 18.60, preprint, 2026, Theorem 1: for an F_2-space V of dimension at least 4, no K ≤ GL(V) transitive on V ∖ {0} has a point stabilizer preserving a complement W_v to F_2 v and transitive on W_v ∖ {0}; Corollary 2: the nonexistence conjecture of Problem 18.60 holds; the Kourovka Notebook (21st issue) marks 18.60 solved, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka_18_59-18_60_solution.pdf.

Uses 1

Statement usesKourovka Problem 18.60 (field-like transitive automorphism groups)

2 fully proved1 ready to prove

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