Localization of nilpotent groups

Definition\(p\)-local groups#✓fully proved

Let \(p\) be a natural number. A group \(H\) is \(p\)-local when, for every natural number \(n\) coprime to \(p\), the map \(x \mapsto x^n\) is a bijection of \(H\).

This is the notion of a \(P\)-local group of Hilton, Mislin and Roitberg, for \(P\) the set whose only element is \(p\).

def IsPLocal (p : ℕ) (H : Type*) [Group H] : Prop :=
  ∀ n : ℕ, Nat.Coprime n p → Function.Bijective fun x : H => x ^ n

Used by 1

Used by\(p\)-localization of a group

Definition\(p\)-localization of a group#✓fully proved

Let \(p\) be a natural number. A homomorphism of groups \(e \colon G \to L\) is a \(p\)-localization when \(L\) is nilpotent and \(p\)-local, and every homomorphism \(f\) from \(G\) to a nilpotent \(p\)-local group \(H\) factors uniquely through \(e\): there is exactly one homomorphism \(g \colon L \to H\) with \(g \circ e = f\).

def IsPLocalization (p : ℕ) {G L : Type} [Group G] [Group L] (e : G →* L) : Prop :=
  Group.IsNilpotent L ∧ IsPLocal p L ∧
    ∀ (H : Type) [Group H], Group.IsNilpotent H → IsPLocal p H →
      ∀ f : G →* H, ∃! g : L →* H, g.comp e = f

Uses 1 · Used by 1

Statement uses\(p\)-local groups

Used bySplitting at a prime

DefinitionSplitting at a prime#✓fully proved

Let \(p\) be a natural number and \(\pi \colon G \to Q\) a homomorphism of groups (in the intended use, the quotient map of an extension). Then \(\pi\) splits at \(p\) when there exist \(p\)-localizations \(e_G \colon G \to G_p\) and \(e_Q \colon Q \to Q_p\) and homomorphisms \(\pi_p \colon G_p \to Q_p\) and \(s \colon Q_p \to G_p\) with \(\pi_p \circ e_G = e_Q \circ \pi\) and \(\pi_p \circ s = \mathrm{id}_{Q_p}\); that is, the localized quotient map \(\pi_p\) has a homomorphic section.

Since \(Q_p\) is nilpotent and \(p\)-local, \(\pi_p\) is the unique homomorphism with \(\pi_p \circ e_G = e_Q \circ \pi\); since localizations are unique up to unique isomorphism, the condition does not depend on the chosen localizations.

def SplitsAtPrime (p : ℕ) {G Q : Type} [Group G] [Group Q] (π : G →* Q) : Prop :=
  ∃ (Gp Qp : Type) (_ : Group Gp) (_ : Group Qp) (eG : G →* Gp) (eQ : Q →* Qp)
    (πp : Gp →* Qp) (s : Qp →* Gp), IsPLocalization p eG ∧ IsPLocalization p eQ ∧
      πp.comp eG = eQ.comp π ∧ πp.comp s = MonoidHom.id Qp

Uses 1 · Used by 1

Statement uses\(p\)-localization of a group

Used byHilton's splitting question for nilpotent extensions (Kourovka 12.93)

DefinitionHilton's splitting question for nilpotent extensions (Kourovka 12.93)#✓fully proved

The proposition that every extension of nilpotent groups with finitely generated quotient that splits at every prime splits: for all nilpotent groups \(N\), \(G\) and \(Q\) with \(Q\) finitely generated, every injective homomorphism \(i \colon N \to G\) and every surjective homomorphism \(\pi \colon G \to Q\) with \(\operatorname{im} i = \ker \pi\) such that the extension splits at every prime \(p\), there is a homomorphism \(s \colon Q \to G\) with \(\pi \circ s = \mathrm{id}_Q\).

def nilpotentExtensionSplitsIfSplitsAtPrimes : Prop :=
  ∀ (N G Q : Type) [Group N] [Group G] [Group Q], Group.IsNilpotent N → Group.IsNilpotent G →
    Group.IsNilpotent Q → Group.FG Q → ∀ (i : N →* G) (π : G →* Q),
      Function.Injective i → Function.Surjective π → i.range = π.ker →
      (∀ p : ℕ, p.Prime → SplitsAtPrime p π) → ∃ s : Q →* G, π.comp s = MonoidHom.id Q

Uses 1 · Used by 1

Statement usesSplitting at a prime

Used byNilpotent extensions that split at every prime need not split

TheoremNilpotent extensions that split at every prime need not split#●ready to prove

It is not true that every extension of nilpotent groups with finitely generated quotient that splits at every prime splits: there are nilpotent groups \(N\), \(G\) and \(Q\) with \(Q\) finitely generated, an injective homomorphism \(i \colon N \to G\) and a surjective homomorphism \(\pi \colon G \to Q\) with \(\operatorname{im} i = \ker \pi\), such that the extension splits at every prime \(p\) (for some \(p\)-localizations \(G_p\) of \(G\) and \(Q_p\) of \(Q\), the localized quotient map \(\pi_p \colon G_p \to Q_p\) has a homomorphic section) but there is no homomorphism \(s \colon Q \to G\) with \(\pi \circ s = \mathrm{id}_Q\).

Lorensen constructed such extensions in which \(N\), \(G\) and \(Q\) are finitely generated and torsion-free.

public theorem_wanted nilpotentExtensionSplitsIfSplitsAtPrimes_refuted :
    ¬ nilpotentExtensionSplitsIfSplitsAtPrimes

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 12.93 (P. Hilton); resolution: K. Lorensen, Nonsplit nilpotent group extensions that split at every prime, Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 2, 213-215, doi:10.1017/S0305004197002211, introduction (first page, shown on Cambridge Core): examples of nilpotent group extensions which split at every p-localization but fail to split, disproving Hilton's conjecture that any nilpotent group extension with finitely generated quotient which splits at every prime must split; in both families of counterexamples the groups are torsion-free and finitely generated. Kourovka Archive, Problem 12.93: 'No, not always', https://doi.org/10.1017/S0305004197002211.

Uses 1

Statement usesHilton's splitting question for nilpotent extensions (Kourovka 12.93)

4 fully proved1 ready to prove

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