Prove2Me

Local inclusion with a pronilpotent normal supplement

Proved
LocalConjugacy.Proof.LocalConjugacy.pronilpotent_local_inclusion

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

group-theorylocal-conjugacy-prosolvableprofinite-groups

Let GG be a profinite group, let N⊴GN\trianglelefteq G be closed and pronilpotent, and let H,K≤GH,K\le G be closed. Assume that GG is prosupersolvable or G/NG/N is pronilpotent. Suppose also that NH=NK=GNH=NK=G and that N∩HN\cap H is normal in NN. If, for every prime pp, some Sylow pro-pp subgroup of KK has a GG-conjugate contained in HH, then

∃g∈G,gKg−1≤H.\exists g\in G,\qquad gKg^{-1}\le H.

This gives local-to-global containment for supplements whose intersection with the pronilpotent normal subgroup satisfies the stated normality condition.

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.pronilpotent_local_inclusion :
∀ {G : Type u_1} [inst : Group.{u_1} G] [inst_1 : TopologicalSpace.{u_1} G]
  [@LocalConjugacy.Proof.LocalConjugacy.Profinite.{u_1} G inst inst_1] (N H K : @Subgroup.{u_1} G inst)
  [inst_3 : @Subgroup.Normal.{u_1} G inst N]
  (hN :
    @IsClosed.{u_1} G inst_1
      (@SetLike.coe.{u_1, u_1} (@Subgroup.{u_1} G inst) G (@Subgroup.instSetLike.{u_1} G inst) N))
  (hH :
    @IsClosed.{u_1} G inst_1
      (@SetLike.coe.{u_1, u_1} (@Subgroup.{u_1} G inst) G (@Subgroup.instSetLike.{u_1} G inst) H))
  (hK :
    @IsClosed.{u_1} G inst_1
      (@SetLike.coe.{u_1, u_1} (@Subgroup.{u_1} G inst) G (@Subgroup.instSetLike.{u_1} G inst) K))
  (hpron :
    @LocalConjugacy.Proof.LocalConjugacy.Pronilpotent.{u_1}
      (@Subtype.{u_1 + 1} G fun (x : G) =>
        @Membership.mem.{u_1, u_1} G (@Subgroup.{u_1} G inst)
          (@SetLike.instMembership.{u_1, u_1} (@Subgroup.{u_1} G inst) G (@Subgroup.instSetLike.{u_1} G inst)) N x)
      (@Subgroup.toGroup.{u_1} G inst N)
      (@instTopologicalSpaceSubtype.{u_1} G
        (fun (x : G) =>
          @Membership.mem.{u_1, u_1} G (@Subgroup.{u_1} G inst)
            (@SetLike.instMembership.{u_1, u_1} (@Subgroup.{u_1} G inst) G (@Subgroup.instSetLike.{u_1} G inst)) N x)
        inst_1))
  (hcase :
    Or (@LocalConjugacy.Proof.LocalConjugacy.Prosupersolvable.{u_1} G inst inst_1)
      (@LocalConjugacy.Proof.LocalConjugacy.Pronilpotent.{u_1}
        (@HasQuotient.Quotient.{u_1, u_1} G (@Subgroup.{u_1} G inst)
          (@QuotientGroup.instHasQuotientSubgroup.{u_1} G inst) N)
        (@QuotientGroup.Quotient.group.{u_1} G inst N inst_3)
        (@QuotientGroup.instTopologicalSpace.{u_1} G inst_1 inst N)))
  (hsH : @LocalConjugacy.Proof.LocalConjugacy.Supplements.{u_1} G inst N H)
  (hsK : @LocalConjugacy.Proof.LocalConjugacy.Supplements.{u_1} G inst N K)
  (hnormal : @LocalConjugacy.Proof.LocalConjugacy.IntersectionNormal.{u_1} G inst N H)
  (hloc : @LocalConjugacy.Proof.LocalConjugacy.LocallyContains.{u_1} G inst inst_1 H K),
  @Exists.{u_1 + 1} G fun (g : G) =>
    @LE.le.{u_1} (@Subgroup.{u_1} G inst)
      (@Preorder.toLE.{u_1} (@Subgroup.{u_1} G inst)
        (@PartialOrder.toPreorder.{u_1} (@Subgroup.{u_1} G inst) (@Subgroup.instPartialOrder.{u_1} G inst)))
      (@LocalConjugacy.Proof.LocalConjugacy.conjugate.{u_1} G inst g K) H := by sorry
Source
Michael C. Burkhart, Local conjugacy in prosolvable groups, https://arxiv.org/abs/2609.37678; supporting formalization lemma, LocalConjugacy/LocalInclusion.lean, lines 49–79; source SHA-256 27de0634721acf5023e8448f725cb3b3e25d40e1a6c0883185f68887e9d4a3af.

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