Prove2Me

Local containment in a complement gives local conjugacy

Proved
LocalConjugacy.Proof.LocalConjugacy.locallyConjugate_of_locallyContains_complement

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

complement-subgroupsgroup-theorylocal-conjugacylocal-conjugacy-prosolvableprofinite-groups

Let GG be a profinite group, let N⊴GN\trianglelefteq G be closed, and let H,K≤GH,K\le G be closed subgroups. Suppose G=NHG=NH, N∩H=1N\cap H=1, and G=NKG=NK. Assume that for every prime pp, the subgroup HH contains a GG-conjugate of some Sylow pro-pp subgroup of KK. Then

∀p prime,∃P∈Syl⁡ppro(K), Q∈Syl⁡ppro(H), g∈G,gPg−1=Q.\forall p\text{ prime},\quad\exists P\in\operatorname{Syl}^{\mathrm{pro}}_p(K),\ Q\in\operatorname{Syl}^{\mathrm{pro}}_p(H),\ g\in G,\qquad gPg^{-1}=Q.

Thus KK and HH are locally conjugate. This strengthens local containment when the containing subgroup is a complement and the other subgroup is a supplement to the same normal subgroup.

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.locallyConjugate_of_locallyContains_complement :
∀ {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)
  [@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))
  (hc : @Subgroup.IsComplement'.{u_1} G inst N H)
  (hsK : @LocalConjugacy.Proof.LocalConjugacy.Supplements.{u_1} G inst N K)
  (hloc : @LocalConjugacy.Proof.LocalConjugacy.LocallyContains.{u_1} G inst inst_1 H K),
  @LocalConjugacy.Proof.LocalConjugacy.LocallyConjugate.{u_1} G inst inst_1 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 16–47; source SHA-256 27de0634721acf5023e8448f725cb3b3e25d40e1a6c0883185f68887e9d4a3af.

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