Permutation groups isomorphic to the rationals

DefinitionSuchkov's question on displacements in permutation groups isomorphic to the rationals (Kourovka 20.98)#✓fully proved

Suchkov's question on displacements in permutation groups isomorphic to the rationals (Kourovka 20.98). The statement that for every subgroup \(G\) of the group of permutations of \(\mathbb{N} = \{0, 1, 2, \dots\}\) that is isomorphic to the additive group \((\mathbb{Q}, +)\), there is an element \(g \in G\) such that the set of displacements \(\{a - g(a) : a \in \mathbb{N}\}\), computed in \(\mathbb{Z}\), is infinite.

Kourovka 20.98 (N. M. Suchkov) is stated for permutations of the positive integers; shifting by one does not change the displacements.

def rationalPermutationGroupInfiniteDisplacement : Prop :=
  ∀ G : Subgroup (Equiv.Perm ℕ), Nonempty (G ≃* Multiplicative ℚ) →
    ∃ g ∈ G, {d : ℤ | ∃ a : ℕ, (a : ℤ) - (g a : ℤ) = d}.Infinite

Used by 1

Used byPermutation groups isomorphic to the rationals have infinite displacement (Kourovka 20.98)

TheoremPermutation groups isomorphic to the rationals have infinite displacement (Kourovka 20.98)#●ready to prove

For every subgroup \(G\) of the group of permutations of \(\mathbb{N} = \{0, 1, 2, \dots\}\) that is isomorphic to the additive group \((\mathbb{Q}, +)\), there is an element \(g \in G\) such that the set of displacements \(\{a - g(a) : a \in \mathbb{N}\}\), computed in \(\mathbb{Z}\), is infinite. That is, the answer to Kourovka Problem 20.98 is yes.

The 21st issue of the Kourovka Notebook marks 20.98 solved ("Yes, it must"), citing Suchkov, Shlepkin and Taysnyov (2024).

public theorem_wanted rationalPermutationGroupInfiniteDisplacement_holds :
    rationalPermutationGroupInfiniteDisplacement

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 20.98 (N. M. Suchkov); resolution: N. M. Suchkov, A. A. Shlepkin and D. A. Taysnyov, On the existence of countable unlimited groups, Siberian Math. J. 65 (2024), no. 6, 1390-1394, abstract: the additive group of the rationals is not isomorphic to a subgroup of the group of all limited permutations of the naturals; Kourovka Notebook (21st issue), Problem 20.98: Yes, it must, https://www.mathnet.ru/eng/smj7921.

Uses 1

Statement usesSuchkov's question on displacements in permutation groups isomorphic to the rationals (Kourovka 20.98)

1 fully proved1 ready to prove

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