One-relator groups as automorphism groups

DefinitionKourovka Problem 16.75: one-relator automorphism groups#✓fully proved

The proposition that there exist a natural number \(n\) and an element \(r\) of the free group on generators \(x_0, \dots, x_{n-1}\) such that the one-relator group \(\langle x_0, \dots, x_{n-1} \mid r \rangle\) is non-abelian and is isomorphic to the automorphism group \(\operatorname{Aut}(H)\) of some group \(H\). The problem is due to M. V. Neshchadim.

def oneRelatorGroupIsAutomorphismGroup : Prop :=
  ∃ (n : ℕ) (r : FreeGroup (Fin n)),
    (∃ a b : PresentedGroup ({r} : Set (FreeGroup (Fin n))), a * b ≠ b * a) ∧
    ∃ (H : Type) (_ : Group H),
      Nonempty (PresentedGroup ({r} : Set (FreeGroup (Fin n))) ≃* MulAut H)

Used by 1

Used bySome non-abelian one-relator group is an automorphism group

TheoremSome non-abelian one-relator group is an automorphism group#●ready to prove

There exist a natural number \(n\) and an element \(r\) of the free group on generators \(x_0, \dots, x_{n-1}\) such that the one-relator group \(\langle x_0, \dots, x_{n-1} \mid r \rangle\) is non-abelian and is isomorphic to the automorphism group \(\operatorname{Aut}(H)\) of some group \(H\). This answers Kourovka Problem 16.75 in the affirmative.

public theorem_wanted oneRelatorGroupIsAutomorphismGroup_holds : oneRelatorGroupIsAutomorphismGroup

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 16.75 (M. V. Neshchadim); resolution: D. J. Collins, Proc. London Math. Soc. (3) 36 (1978), no. 3, 480-493; V. A. Churkin, letter of 5 April 2006, as recorded in the Kourovka Archive, Kourovka Archive, Problem 16.75 (Yes, it can: for integers r, s with gcd(r, s) = 1, |r| ≠ |s| and r - s even, G = ⟨a, b | a⁻¹ b^r a = b^s⟩ is isomorphic to Aut(Aut(G)), Collins).

Uses 1

Statement usesKourovka Problem 16.75: one-relator automorphism groups

1 fully proved1 ready to prove

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