Groups acting freely on abelian groups

DefinitionKourovka Problem 16.54 (free actions and embeddings in skew-fields)#✓fully proved

The statement, a question of V. D. Mazurov, that every group \(G\) with a free action by automorphisms on a nontrivial abelian group \(V\), meaning \(g \cdot v \ne v\) for all \(g \ne 1\) in \(G\) and all \(v \ne 0\) in \(V\), has an injective homomorphism into the multiplicative group of some division ring.

def freeActionEmbedsInSkewField : Prop :=
  ∀ (G : Type) [Group G],
    (∃ (V : Type) (_ : AddCommGroup V) (_ : Nontrivial V) (_ : DistribMulAction G V),
      ∀ g : G, g ≠ 1 → ∀ v : V, v ≠ 0 → g • v ≠ v) →
    ∃ (D : Type) (_ : DivisionRing D) (f : G →* Dˣ), Function.Injective f

Used by 1

Used byGroups acting freely on abelian groups need not embed in skew-fields (Kourovka 16.54)

TheoremGroups acting freely on abelian groups need not embed in skew-fields (Kourovka 16.54)#●ready to prove

There is a group \(G\) with an action by automorphisms on a nontrivial abelian group \(V\) such that \(g \cdot v \ne v\) for all \(g \ne 1\) in \(G\) and all \(v \ne 0\) in \(V\), but with no injective homomorphism into the multiplicative group of any division ring. This answers Kourovka Problem 16.54 in the negative.

The Kourovka Archive answers "No, it is not" (Nedrenko): the group \(2.A_5.2\) with a quaternion Sylow \(2\)-subgroup acts freely on an elementary abelian group of order \(7^4\), but by Theorem 7 of Amitsur's classification of finite subgroups of division rings it does not embed in the multiplicative group of any skew-field.

public theorem_wanted freeActionEmbedsInSkewField_refuted : ¬ freeActionEmbedsInSkewField

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 16.54 (V. D. Mazurov); resolution: D. Nedrenko, letter of 20 January 2014, as recorded in the Kourovka Archive; S. A. Amitsur, Finite subgroups of division rings, Trans. Amer. Math. Soc. 80 (1955), 361-386, Kourovka Archive, Problem 16.54 (No: 2.A_5.2 with a quaternion Sylow 2-subgroup acts freely on an elementary abelian group of order 7^4 but does not embed in the multiplicative group of a skew-field, by Amitsur's Theorem 7).

Uses 1

Statement usesKourovka Problem 16.54 (free actions and embeddings in skew-fields)

1 fully proved1 ready to prove

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