A normal Sylow lies below a different prime-index subgroup

Proved

LocalConjugacy.Proof.LocalConjugacy.prosupersolvable_prime_index_above_normal_sylow

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

group-theorylocal-conjugacy-prosolvableprofinite-groupssupersolvable-groupssylow-theory

Let $G$ be a prosupersolvable profinite group, let $p$ be prime, and let $P\trianglelefteq G$ be a proper Sylow pro- $p$ subgroup. Then

\exists K\trianglelefteq_{\mathrm{open}}G,\qquad P\le K,\quad [G:K]\text{ is prime},\quad [G:K]\ne p.

Here prosupersolvability requires every finite continuous quotient to be supersolvable. This provides the prime-index reduction above a proper normal Sylow 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.prosupersolvable_prime_index_above_normal_sylow :
∀ {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]
  (hG : @LocalConjugacy.Proof.LocalConjugacy.Prosupersolvable.{u_1} G inst inst_1) {p : Nat} [Fact (Nat.Prime p)]
  (P : @Subgroup.{u_1} G inst) [@Subgroup.Normal.{u_1} G inst P]
  (hP :
    @LocalConjugacy.Proof.LocalConjugacy.IsSylowPro.{u_1} p G inst inst_1
      (@Top.top.{u_1} (@Subgroup.{u_1} G inst) (@Subgroup.instTop.{u_1} G inst)) P)
  (hproper :
    @Ne.{u_1 + 1} (@Subgroup.{u_1} G inst) P
      (@Top.top.{u_1} (@Subgroup.{u_1} G inst) (@Subgroup.instTop.{u_1} G inst))),
  @Exists.{u_1 + 1} (@OpenNormalSubgroup.{u_1} G inst inst_1) fun (K : @OpenNormalSubgroup.{u_1} G inst inst_1) =>
    And
      (@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)))
        P (@OpenSubgroup.toSubgroup.{u_1} G inst inst_1 (@OpenNormalSubgroup.toOpenSubgroup.{u_1} G inst inst_1 K)))
      (And
        (Nat.Prime
          (@Subgroup.index.{u_1} G inst
            (@OpenSubgroup.toSubgroup.{u_1} G inst inst_1 (@OpenNormalSubgroup.toOpenSubgroup.{u_1} G inst inst_1 K))))
        (@Ne.{1} Nat
          (@Subgroup.index.{u_1} G inst
            (@OpenSubgroup.toSubgroup.{u_1} G inst inst_1 (@OpenNormalSubgroup.toOpenSubgroup.{u_1} G inst inst_1 K)))
          p)) := by sorry

Source

Michael C. Burkhart, Local conjugacy in prosolvable groups, https://arxiv.org/abs/2609.37678; supporting formalization lemma, LocalConjugacy/SupersolvableReductions.lean, lines 154–185; source SHA-256 29509bd9dc03344a0acef30e2bd052c64781f5f1093f7c226e23a6aa0ad6969d.

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