Linear groups inside division rings

DefinitionKourovka Problem 11.110 (B. Hartley)#✓fully proved

The group \(\mathrm{SL}(2, \mathbb{Q})\) embeds in the multiplicative group of some division ring: there are a division ring \(D\) and an injective group homomorphism \(\mathrm{SL}(2, \mathbb{Q}) \to D^\times\).

def specialLinearRationalEmbedsInDivisionRing : Prop :=
  ∃ (D : Type) (_ : DivisionRing D) (f : Matrix.SpecialLinearGroup (Fin 2) ℚ →* Dˣ),
    Function.Injective f

Used by 1

Used bySL(2, ℚ) does not embed in a division ring

TheoremSL(2, ℚ) does not embed in a division ring#●ready to prove

There is no division ring \(D\) with an injective group homomorphism \(\mathrm{SL}(2, \mathbb{Q}) \to D^\times\): the group \(\mathrm{SL}(2, \mathbb{Q})\) does not embed in the multiplicative group of any division ring.

public theorem_wanted specialLinearRationalEmbedsInDivisionRing_refuted :
    ¬ specialLinearRationalEmbedsInDivisionRing

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 11.110 (B. Hartley); resolution: W. Dicks and B. Hartley, Commun. Algebra 19 (1991), 1919-1943, Kourovka Archive, Problem 11.110 ("No, it is not").

Uses 1

Statement usesKourovka Problem 11.110 (B. Hartley)

1 fully proved1 ready to prove

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