Subgroups of the infinite symmetric group

TheoremThe additive group of the reals embeds in the infinite symmetric group#●ready to prove

There is an injective group homomorphism from the additive group \((\mathbb{R}, +)\) of real numbers into the group \(S_\infty\) of all permutations of the natural numbers.

theorem_wanted real_addGroup_embeds_in_perm_nat :
    ∃ f : Multiplicative ℝ →* Equiv.Perm ℕ, Function.Injective f

Sources

• J. Schreier and S. Ulam, Problem 95 of the Scottish Book; affirmative answer of Schreier and Ulam, November 1935, recorded in the typescript addendum Ad 95. Multiplicative ℝ is the additive group ℝ viewed multiplicatively, so a monoid homomorphism out of it is an additive homomorphism out of ℝ.

DefinitionClaim of Kourovka Problem 18.36#✓fully proved

The claim (J. D. Dixon) that every group \(G\) that admits a metrizable topology making it a separable topological group admits an injective group homomorphism into the group \(\mathrm{Sym}(\mathbb{N})\) of all permutations of the natural numbers.

def separableMetrizableGroupEmbedsInSym : Prop :=
  ∀ (G : Type) [Group G],
    (∃ t : TopologicalSpace G, @IsTopologicalGroup G t _ ∧
      @TopologicalSpace.MetrizableSpace G t ∧ @TopologicalSpace.SeparableSpace G t) →
    ∃ f : G →* Equiv.Perm ℕ, Function.Injective f

Used by 1

Used bySeparable metrizable groups need not embed in the infinite symmetric group

TheoremSeparable metrizable groups need not embed in the infinite symmetric group#●ready to prove

The claim of Kourovka Problem 18.36 is false: there is a group \(G\) that admits a metrizable topology making it a separable topological group but admits no injective group homomorphism into the group \(\mathrm{Sym}(\mathbb{N})\) of all permutations of the natural numbers.

Rosendal and Solecki (2007) give an uncountable connected Polish group (a separable completely metrizable topological group) every abstract homomorphism of which into \(\mathrm{Sym}(\mathbb{N})\) is trivial, so it has no injective one.

public theorem_wanted separableMetrizableGroupEmbedsInSym_refuted :
    ¬ separableMetrizableGroupEmbedsInSym

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Problem 18.36 (J. D. Dixon); resolution: C. Rosendal and S. Solecki, Israel J. Math. 162 (2007), 349-371, an uncountable connected Polish group whose every abstract homomorphism into Sym(ℕ) is trivial; see also A. Kaïchouh and F. Le Maître, Bull. London Math. Soc. 47 (2015), 996-1009 (S. Corson, letter of 5 June 2024); the Kourovka Notebook (21st issue) marks 18.36 solved, https://arxiv.org/abs/1401.0300v49.

Uses 1

Statement usesClaim of Kourovka Problem 18.36

1 fully proved2 ready to prove

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