Separability of isolated subgroups

Definition\(p'\)-isolated subgroup#✓fully proved

Let \(p\) be a natural number and let \(H\) be a subgroup of a group \(F\). Then \(H\) is \(p'\)-isolated if, for every \(x \in F\) and every natural number \(n > 0\) that is not divisible by \(p\), \(x^n \in H\) implies \(x \in H\).

def IsPPrimeIsolated (p : ℕ) {F : Type*} [Group F] (H : Subgroup F) : Prop :=
  ∀ (x : F) (n : ℕ), 0 < n → ¬ p ∣ n → x ^ n ∈ H → x ∈ H

Used by 1

Used byMoldavanskiĭ's question on \(p\)-separability of isolated subgroups (Kourovka 15.60)

DefinitionSeparability in the class of finite \(p\)-groups#✓fully proved

Let \(p\) be a natural number and let \(H\) be a subgroup of a group \(F\). Then \(H\) is separable in the class of finite \(p\)-groups if for every \(x \in F\) with \(x \notin H\) there are a finite \(p\)-group \(Q\) (a finite group in which every element \(g\) satisfies \(g^{p^k} = 1\) for some natural number \(k\)) and a group homomorphism \(f \colon F \to Q\) with \(f(x) \notin f(H)\).

def IsPSeparable (p : ℕ) {F : Type*} [Group F] (H : Subgroup F) : Prop :=
  ∀ x : F, x ∉ H → ∃ (Q : Type) (_ : Group Q) (_ : Finite Q), IsPGroup p Q ∧
    ∃ f : F →* Q, f x ∉ H.map f

Used by 1

Used byMoldavanskiĭ's question on \(p\)-separability of isolated subgroups (Kourovka 15.60)

DefinitionMoldavanskiĭ's question on \(p\)-separability of isolated subgroups (Kourovka 15.60)#✓fully proved

The statement that there is a prime \(p\) such that, for every set \(\alpha\), every finitely generated \(p'\)-isolated subgroup of the free group on \(\alpha\) is separable in the class of finite \(p\)-groups.

def isolatedFGSubgroupsPSeparable : Prop :=
  ∃ p : ℕ, p.Prime ∧ ∀ (α : Type) (H : Subgroup (FreeGroup α)), H.FG → IsPPrimeIsolated p H →
    IsPSeparable p H

Uses 2 · Used by 1

Statement uses\(p'\)-isolated subgroup · Separability in the class of finite \(p\)-groups

Used byIsolated subgroups of free groups need not be \(p\)-separable

TheoremIsolated subgroups of free groups need not be \(p\)-separable#●ready to prove

For every prime \(p\) there are a set \(\alpha\) and a finitely generated \(p'\)-isolated subgroup of the free group on \(\alpha\) that is not separable in the class of finite \(p\)-groups. That is, the statement of Kourovka Problem 15.60 fails for every prime \(p\).

Bardakov proved that every nonabelian free group has a finitely generated isolated subgroup \(H\) (\(x^n \in H\) with \(n \ne 0\) implies \(x \in H\)) that is not separable in the class of nilpotent groups; it is \(p'\)-isolated for every prime \(p\), and finite \(p\)-groups are nilpotent. The Kourovka Archive answers 15.60 "No, it is not".

public theorem_wanted isolatedFGSubgroupsPSeparable_refuted : ¬ isolatedFGSubgroupsPSeparable

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 15.60 (D. I. Moldavanskiĭ); resolution: V. G. Bardakov, On D. I. Moldavanskii's question about p-separable subgroups of a free group, Siberian Math. J. 45 (2004), no. 3, 416-419, arXiv:math/0404292, abstract (every nonabelian free group has a finitely generated isolated subgroup that is not separable in the class of nilpotent groups); Kourovka Archive, Problem 15.60 (answer: No, it is not), https://arxiv.org/abs/math/0404292.

Uses 1

Statement usesMoldavanskiĭ's question on \(p\)-separability of isolated subgroups (Kourovka 15.60)

3 fully proved1 ready to prove

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