Prove2Me

Extending stable Sylow cocycles in the supersolvable case

Proved
LocalConjugacy.Proof.LocalConjugacy.supersolvable_sylow_extension

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

group-cohomologygroup-theorylocal-conjugacy-prosolvablesupersolvable-groups

Let JJ be a profinite group acting continuously by automorphisms on a finite discrete pp-group NN, where pp is prime. Assume that the semidirect product N⋊JN\rtimes J, with its product topology, is prosupersolvable. Let PP be a Sylow pro-pp subgroup of JJ, and let f:P→Nf:P\to N be a continuous cocycle. Suppose ff is JJ-stable: for every j∈Jj\in J, the cocycle x↦j⋅f(j−1xj)x\mapsto j\cdot f(j^{-1}xj) is cohomologous to ff on P∩jPj−1P\cap jPj^{-1}. Then

∃F∈Zcts1(J,N), ∃n∈N, ∀x∈P,f(x)=n−1F(x)(x⋅n).\exists F\in Z^1_{\mathrm{cts}}(J,N),\ \exists n\in N,\ \forall x\in P,\qquad f(x)=n^{-1}F(x)(x\cdot n).

Here Zcts1(J,N)Z^1_{\mathrm{cts}}(J,N) denotes continuous nonabelian cocycles, and cohomology uses the displayed change by a fixed coefficient. This establishes extension of stable Sylow classes in the supersolvable semidirect-product case.

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.supersolvable_sylow_extension :
∀ {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]
  (hG :
    @LocalConjugacy.Proof.LocalConjugacy.Prosupersolvable.{max u_2 u_1}
      (@LocalConjugacy.Proof.LocalConjugacy.ActionProduct.{u_1, u_2} J N inst inst_1 inst_7)
      (@SemidirectProduct.instGroup.{u_2, u_1} N J inst_1 inst
        (@MulDistribMulAction.toMulAut.{u_1, u_2} J N inst
          (@DivInvMonoid.toMonoid.{u_2} N (@Group.toDivInvMonoid.{u_2} N inst_1)) inst_7))
      (@LocalConjugacy.Proof.LocalConjugacy.semidirectTopology.{u_1, u_2} J N inst inst_1 inst_2 inst_4
        (@MulDistribMulAction.toMulAut.{u_1, u_2} J N inst
          (@DivInvMonoid.toMonoid.{u_2} N (@Group.toDivInvMonoid.{u_2} N inst_1)) inst_7)))
  {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)
  (f : @LocalConjugacy.Proof.LocalConjugacy.Cocycle.{u_1, u_2} J N inst inst_1 inst_2 inst_4 inst_7 P)
  (hf :
    @LocalConjugacy.Proof.LocalConjugacy.InvariantUnder.{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)) P f),
  @LocalConjugacy.Proof.LocalConjugacy.Cocycle.ExtendsTo.{u_1, u_2} J N inst inst_1 inst_2 inst_4 inst_7 P f
    (@Top.top.{u_1} (@Subgroup.{u_1} J inst) (@Subgroup.instTop.{u_1} J inst)) := by sorry
Source
Michael C. Burkhart, Local conjugacy in prosolvable groups, https://arxiv.org/abs/2609.37678; supporting formalization lemma, LocalConjugacy/SupersolvableRestriction.lean, lines 105–159; source SHA-256 7e610167f4cb688259c0ca09f4281f15dff890c6344f3ca38343d1c9135a00a7.

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