Extensions of the additive group of a division ring

DefinitionPalyutin's splitting problem for division rings (Kourovka 18.76)#✓fully proved

The statement that for every division ring \(A\), every subgroup \(G\) of its unit group \(A^\times\) and every group \(E\), if \(i\) is an injective homomorphism from the additive group of \(A\) to \(E\) and \(\pi : E \to G\) is a surjective homomorphism whose kernel is the image of \(i\), such that \(e \, i(a) \, e^{-1} = i(\pi(e) a)\) for all \(e \in E\) and \(a \in A\) (so \(G\) acts on the kernel by left multiplication), then the extension splits: there is a homomorphism \(s : G \to E\) with \(\pi(s(g)) = g\) for every \(g \in G\).

def divisionRingMultiplicationExtensionsSplit : Prop :=
  ∀ (A : Type) [DivisionRing A] (G : Subgroup Aˣ) (E : Type) [Group E]
    (i : Multiplicative A →* E) (π : E →* G), Function.Injective i → Function.Surjective π →
    i.range = π.ker →
    (∀ (e : E) (a : A), e * i (Multiplicative.ofAdd a) * e⁻¹ =
      i (Multiplicative.ofAdd (((π e : Aˣ) : A) * a))) →
    ∃ s : G →* E, ∀ g : G, π (s g) = g

Used by 1

Used byExtensions of a division ring's additive group by units need not split (Kourovka 18.76)

TheoremExtensions of a division ring's additive group by units need not split (Kourovka 18.76)#●ready to prove

There exist a division ring \(A\), a subgroup \(G\) of its unit group \(A^\times\), a group \(E\), an injective homomorphism \(i\) from the additive group of \(A\) to \(E\) and a surjective homomorphism \(\pi : E \to G\) whose kernel is the image of \(i\), with \(e \, i(a) \, e^{-1} = i(\pi(e) a)\) for all \(e \in E\) and \(a \in A\), such that no homomorphism \(s : G \to E\) satisfies \(\pi(s(g)) = g\) for all \(g \in G\). That is, the splitting statement of Kourovka Problem 18.76 is false.

public theorem_wanted divisionRingMultiplicationExtensionsSplit_refuted :
    ¬ divisionRingMultiplicationExtensionsSplit

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 18.76 (E. A. Palyutin); resolution: M. Kinyon and J. Urban, Results on problems from the Kourovka Notebook, preprint, version 4, 19 September 2026; Achyuth Jayadevan, preprint of 14 September 2026, https://zenodo.org/records/22751423, Kinyon-Urban Section 3.28: G = F × L with F, L free of rank 2, D the division closure of kG in a Mal'cev-Neumann division ring (k = Q or a prime field), and a nonsplit extension of (D, +) by G whose conjugation action on the kernel is left multiplication; AI solutions confirmed by B. A. F. Wehrfritz, accepted in the Kourovka Notebook (21st issue), https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka-mathematics.pdf.

Uses 1

Statement usesPalyutin's splitting problem for division rings (Kourovka 18.76)

1 fully proved1 ready to prove

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