Sylow subgroups of pronilpotent profinite groups are normal

Proved

LocalConjugacy.Proof.LocalConjugacy.sylowPro_normal_of_pronilpotent

by burkh4rt · Sep 30, 2026 · Mathlib 0df444a (Lean v4.33.1)

group-theorylocal-conjugacy-prosolvableprofinite-groupspronilpotent-groupssylow-theory

Let $J$ be a profinite group whose finite continuous quotients are all nilpotent. For a prime $p$ , let $P\le J$ be a Sylow pro- $p$ subgroup, that is, a maximal closed pro- $p$ subgroup. Then

P\trianglelefteq J.

This supplies the normality needed for structural reductions involving Sylow subgroups of pronilpotent groups.

Preamble

import Definitions.Def_LocalConjugacy_Groups
import Definitions.Def_LocalConjugacy_Cohomology
import Definitions.Def_LocalConjugacy_Examples
import Definitions.Def_LocalConjugacy_Proof_Definitions
import Definitions.Def_LocalConjugacy_Proof_Bridges
import Definitions.Def_LocalConjugacy_Proof_Counterexamples_Heisenberg
import Definitions.Def_LocalConjugacy_Proof_Counterexamples_HeisenbergStructure
import Definitions.Def_LocalConjugacy_Proof_Counterexamples_HeisenbergSupersolvable
import Definitions.Def_LocalConjugacy_Proof_ConcreteGroups
import Definitions.Def_LocalConjugacy_Targets
import Definitions.Def_LocalConjugacy_Proof_Compactness
import Definitions.Def_LocalConjugacy_Proof_ProfiniteSylow
import Definitions.Def_LocalConjugacy_Proof_StructuralImages
import Definitions.Def_LocalConjugacy_Proof_FiniteAbelianCohomology
import Definitions.Def_LocalConjugacy_Proof_AbelianComplement
import Definitions.Def_LocalConjugacy_Proof_QuotientReduction
import Definitions.Def_LocalConjugacy_Proof_Cohomology
import Definitions.Def_LocalConjugacy_Proof_InvariantRestriction
import Definitions.Def_LocalConjugacy_Proof_CocycleActions
import Definitions.Def_LocalConjugacy_Proof_CoprimeCohomology
import Definitions.Def_LocalConjugacy_Proof_CocycleDescent
import Definitions.Def_LocalConjugacy_Proof_CocycleZorn
import Definitions.Def_LocalConjugacy_Proof_CocycleProducts
import Definitions.Def_LocalConjugacy_Proof_FiniteCoefficientSubgroup
import Definitions.Def_LocalConjugacy_Proof_CocycleInvarianceSubgroup
import Definitions.Def_LocalConjugacy_Proof_CocycleInjectivity
import Definitions.Def_LocalConjugacy_Proof_CocycleRebase
import Definitions.Def_LocalConjugacy_Proof_FiniteHall
import Definitions.Def_LocalConjugacy_Proof_SupersolvableStructure
import Definitions.Def_LocalConjugacy_Proof_ProfiniteHall
import Definitions.Def_LocalConjugacy_Proof_ActionProductTopology
import Definitions.Def_LocalConjugacy_Proof_HallCohomology
import Definitions.Def_LocalConjugacy_Proof_SupersolvableRestriction
import Definitions.Def_LocalConjugacy_Proof_NilpotentCoefficients
import Definitions.Def_LocalConjugacy_Proof_NonabelianComplement
import Definitions.Def_LocalConjugacy_Proof_ComplementSupersolvable
import Definitions.Def_LocalConjugacy_Proof_Counterexamples_Quaternion
import Definitions.Def_LocalConjugacy_Proof_QuaternionCohomology
import Definitions.Def_LocalConjugacy_Proof_QuaternionMatrices
import Definitions.Def_LocalConjugacy_Proof_QuaternionAction
import Definitions.Def_LocalConjugacy_Proof_QuaternionComplements

universe u_1

Formal statement

theorem LocalConjugacy.Proof.LocalConjugacy.sylowPro_normal_of_pronilpotent :
∀ {J : Type u_1} [inst : Group.{u_1} J] [inst_1 : TopologicalSpace.{u_1} J]
  [@LocalConjugacy.Proof.LocalConjugacy.Profinite.{u_1} J inst inst_1]
  (hJ : @LocalConjugacy.Proof.LocalConjugacy.Pronilpotent.{u_1} J inst inst_1) {p : Nat} [Fact (Nat.Prime p)]
  (P : @Subgroup.{u_1} J inst)
  (hP :
    @LocalConjugacy.Proof.LocalConjugacy.IsSylowPro.{u_1} p J inst inst_1
      (@Top.top.{u_1} (@Subgroup.{u_1} J inst) (@Subgroup.instTop.{u_1} J inst)) P),
  @Subgroup.Normal.{u_1} J inst P := by sorry

Source

Michael C. Burkhart, Local conjugacy in prosolvable groups, https://arxiv.org/abs/2609.37678; supporting formalization lemma, LocalConjugacy/PronilpotentSylow.lean, lines 14–31; source SHA-256 45cb1e038f2c2ddffd5cdc21dc870015e7d3ecb0b9e32bfef9fd68788e0c087a.

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