Prove2Me

Local conjugacy of complements with finite pronilpotent kernel

Proved
LocalConjugacy.Proof.LocalConjugacy.finite_complement_conjugacy

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

group-theorylocal-conjugacylocal-conjugacy-prosolvableprofinite-groups

Let GG be profinite, let N⊴GN\trianglelefteq G be closed, finite, discrete in its inherited topology, and pronilpotent, and let H,K≤GH,K\le G be closed complements of NN. Thus G=NH=NKG=NH=NK and N∩H=N∩K={1}N\cap H=N\cap K=\{1\}. Assume either that GG is prosupersolvable or that G/NG/N is pronilpotent. Then

H and K are conjugate in G⟺H and K are locally conjugate in G.H\text{ and }K\text{ are conjugate in }G\quad\Longleftrightarrow\quad H\text{ and }K\text{ are locally conjugate in }G.

Local conjugacy means that, for every prime pp, some Sylow pro-pp subgroup of HH is GG-conjugate to some Sylow pro-pp subgroup of KK.

This is the finite-coefficient complement case used in the passage to the general supplement theorem.

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.finite_complement_conjugacy :
∀ {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]
  [@DiscreteTopology.{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)
      (@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)]
  [Finite.{u_1 + 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)]
  (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)))
  (hHN : @LocalConjugacy.Proof.LocalConjugacy.Supplements.{u_1} G inst N H)
  (hKN : @LocalConjugacy.Proof.LocalConjugacy.Supplements.{u_1} G inst N K)
  (hNH :
    @Eq.{u_1 + 1} (@Subgroup.{u_1} G inst)
      (@Min.min.{u_1} (@Subgroup.{u_1} G inst) (@Subgroup.instMin.{u_1} G inst) N H)
      (@Bot.bot.{u_1} (@Subgroup.{u_1} G inst) (@Subgroup.instBot.{u_1} G inst)))
  (hNK :
    @Eq.{u_1 + 1} (@Subgroup.{u_1} G inst)
      (@Min.min.{u_1} (@Subgroup.{u_1} G inst) (@Subgroup.instMin.{u_1} G inst) N K)
      (@Bot.bot.{u_1} (@Subgroup.{u_1} G inst) (@Subgroup.instBot.{u_1} G inst))),
  Iff (@LocalConjugacy.Proof.LocalConjugacy.Conjugate.{u_1} G inst H K)
    (@LocalConjugacy.Proof.LocalConjugacy.LocallyConjugate.{u_1} G inst inst_1 H K) := by sorry
Source
Michael C. Burkhart, Local conjugacy in prosolvable groups, https://arxiv.org/abs/2609.37678; supporting formalization lemma, LocalConjugacy/ComplementSupersolvable.lean, lines 40–63; source SHA-256 20498d6f205153845353972821b008368a537322d3915950c98ce41b3a8427ac.

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