Prove2Me

Injectivity of Sylow restriction for pronilpotent groups

Proved
LocalConjugacy.Proof.LocalConjugacy.pronilpotent_sylow_restriction_injective_zorn

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

group-cohomologygroup-theorylocal-conjugacy-prosolvableprofinite-groups

Let JJ be a pronilpotent profinite group acting continuously by automorphisms on a finite discrete pp-group NN, where pp is prime. Let PP be a Sylow pro-pp subgroup of JJ, and let φ,ψ:J→N\varphi,\psi:J\to N be continuous nonabelian cocycles. If their restrictions to PP are cohomologous, then

[φ∣P]=[ψ∣P]⟹[φ]=[ψ] in H1(J,N).[\varphi|_P]=[\psi|_P]\quad\Longrightarrow\quad[\varphi]=[\psi]\text{ in }H^1(J,N).

Cohomology means that ψ(x)=n−1φ(x)(x⋅n)\psi(x)=n^{-1}\varphi(x)(x\cdot n) for one fixed n∈Nn\in N throughout the domain. This is the injectivity part of Sylow restriction for pronilpotent acting 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 u_2
Formal statement
theorem LocalConjugacy.Proof.LocalConjugacy.pronilpotent_sylow_restriction_injective_zorn :
∀ {J : Type u_1} {N : Type u_2} [inst : Group.{u_1} J] [inst_1 : Group.{u_2} N] [inst_2 : TopologicalSpace.{u_1} J]
  [@LocalConjugacy.Proof.LocalConjugacy.Profinite.{u_1} J inst inst_2] [inst_4 : TopologicalSpace.{u_2} N]
  [@DiscreteTopology.{u_2} N inst_4] [Finite.{u_2 + 1} N]
  [inst_7 :
    @MulDistribMulAction.{u_1, u_2} J N (@DivInvMonoid.toMonoid.{u_1} J (@Group.toDivInvMonoid.{u_1} J inst))
      (@DivInvMonoid.toMonoid.{u_2} N (@Group.toDivInvMonoid.{u_2} N inst_1))]
  [@ContinuousSMul.{u_1, u_2} J N
      (@SemigroupAction.toSMul.{u_1, u_2} J N
        (@Monoid.toSemigroup.{u_1} J (@DivInvMonoid.toMonoid.{u_1} J (@Group.toDivInvMonoid.{u_1} J inst)))
        (@MulAction.toSemigroupAction.{u_1, u_2} J N
          (@DivInvMonoid.toMonoid.{u_1} J (@Group.toDivInvMonoid.{u_1} J inst))
          (@MulDistribMulAction.toMulAction.{u_1, u_2} J N
            (@DivInvMonoid.toMonoid.{u_1} J (@Group.toDivInvMonoid.{u_1} J inst))
            (@DivInvMonoid.toMonoid.{u_2} N (@Group.toDivInvMonoid.{u_2} N inst_1)) inst_7)))
      inst_2 inst_4]
  (hJ : @LocalConjugacy.Proof.LocalConjugacy.Pronilpotent.{u_1} J inst inst_2) {p : Nat} [Fact (Nat.Prime p)]
  (hN : @IsPGroup.{u_2} p N inst_1) (P : @Subgroup.{u_1} J inst)
  (hP :
    @LocalConjugacy.Proof.LocalConjugacy.IsSylowPro.{u_1} p J inst inst_2
      (@Top.top.{u_1} (@Subgroup.{u_1} J inst) (@Subgroup.instTop.{u_1} J inst)) P)
  (φ ψ :
    @LocalConjugacy.Proof.LocalConjugacy.Cocycle.{u_1, u_2} J N inst inst_1 inst_2 inst_4 inst_7
      (@Top.top.{u_1} (@Subgroup.{u_1} J inst) (@Subgroup.instTop.{u_1} J inst)))
  (hφψ :
    @LocalConjugacy.Proof.LocalConjugacy.Cohomologous.{u_1, u_2} J N inst inst_1 inst_2 inst_4 inst_7 P
      (@LocalConjugacy.Proof.LocalConjugacy.restrictCocycle.{u_1, u_2} J N inst inst_1 inst_2 inst_4 inst_7 P
        (@Top.top.{u_1} (@Subgroup.{u_1} J inst) (@Subgroup.instTop.{u_1} J inst))
        (have this :
          @LE.le.{u_1} (@Subgroup.{u_1} J inst)
            (@Preorder.toLE.{u_1} (@Subgroup.{u_1} J inst)
              (@PartialOrder.toPreorder.{u_1} (@Subgroup.{u_1} J inst) (@Subgroup.instPartialOrder.{u_1} J inst)))
            P (@Top.top.{u_1} (@Subgroup.{u_1} J inst) (@Subgroup.instTop.{u_1} J inst)) :=
          @le_top.{u_1} (@Subgroup.{u_1} J inst)
            (@Preorder.toLE.{u_1} (@Subgroup.{u_1} J inst)
              (@PartialOrder.toPreorder.{u_1} (@Subgroup.{u_1} J inst) (@Subgroup.instPartialOrder.{u_1} J inst)))
            (@BoundedOrder.toOrderTop.{u_1} (@Subgroup.{u_1} J inst)
              (@Preorder.toLE.{u_1} (@Subgroup.{u_1} J inst)
                (@PartialOrder.toPreorder.{u_1} (@Subgroup.{u_1} J inst) (@Subgroup.instPartialOrder.{u_1} J inst)))
              (@CompleteLattice.toBoundedOrder.{u_1} (@Subgroup.{u_1} J inst)
                (@Subgroup.instCompleteLattice.{u_1} J inst)))
            P;
        this)
        φ)
      (@LocalConjugacy.Proof.LocalConjugacy.restrictCocycle.{u_1, u_2} J N inst inst_1 inst_2 inst_4 inst_7 P
        (@Top.top.{u_1} (@Subgroup.{u_1} J inst)
          (@OrderTop.toTop.{u_1} (@Subgroup.{u_1} J inst)
            (@Preorder.toLE.{u_1} (@Subgroup.{u_1} J inst)
              (@PartialOrder.toPreorder.{u_1} (@Subgroup.{u_1} J inst) (@Subgroup.instPartialOrder.{u_1} J inst)))
            (@BoundedOrder.toOrderTop.{u_1} (@Subgroup.{u_1} J inst)
              (@Preorder.toLE.{u_1} (@Subgroup.{u_1} J inst)
                (@PartialOrder.toPreorder.{u_1} (@Subgroup.{u_1} J inst) (@Subgroup.instPartialOrder.{u_1} J inst)))
              (@CompleteLattice.toBoundedOrder.{u_1} (@Subgroup.{u_1} J inst)
                (@Subgroup.instCompleteLattice.{u_1} J inst)))))
        (@le_top.{u_1} (@Subgroup.{u_1} J inst)
          (@Preorder.toLE.{u_1} (@Subgroup.{u_1} J inst)
            (@PartialOrder.toPreorder.{u_1} (@Subgroup.{u_1} J inst) (@Subgroup.instPartialOrder.{u_1} J inst)))
          (@BoundedOrder.toOrderTop.{u_1} (@Subgroup.{u_1} J inst)
            (@Preorder.toLE.{u_1} (@Subgroup.{u_1} J inst)
              (@PartialOrder.toPreorder.{u_1} (@Subgroup.{u_1} J inst) (@Subgroup.instPartialOrder.{u_1} J inst)))
            (@CompleteLattice.toBoundedOrder.{u_1} (@Subgroup.{u_1} J inst)
              (@Subgroup.instCompleteLattice.{u_1} J inst)))
          P)
        ψ)),
  @LocalConjugacy.Proof.LocalConjugacy.Cohomologous.{u_1, u_2} J N inst inst_1 inst_2 inst_4 inst_7
    (@Top.top.{u_1} (@Subgroup.{u_1} J inst) (@Subgroup.instTop.{u_1} J inst)) φ ψ := by sorry
Source
Michael C. Burkhart, Local conjugacy in prosolvable groups, https://arxiv.org/abs/2609.37678; supporting formalization lemma, LocalConjugacy/PronilpotentZorn.lean, lines 62–93; source SHA-256 513161a3df0f10b55069b0bafeda3c91abd1bb6268dcd7b33b0a1ec0764512d3.

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