Normality of the Sylow subgroup at the largest allowed prime

Proved

LocalConjugacy.Proof.LocalConjugacy.supersolvable_largest_sylow_normal

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

group-theorylocal-conjugacy-prosolvablesupersolvable-groupssylow-theory

Let $G$ be a finite supersolvable group, let $p$ be prime, and assume that every prime divisor $r$ of $|G|$ satisfies $r\le p$ . For any Sylow $p$ -subgroup $P$ of $G$ ,

P\trianglelefteq G.

This includes $p\nmid|G|$ , when $P$ is trivial. It isolates the normal Sylow factor at the upper end of a supersolvable group's prime spectrum.

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.supersolvable_largest_sylow_normal :
∀ {G : Type u_1} [inst : Group.{u_1} G] [Finite.{u_1 + 1} G]
  (hG : @LocalConjugacy.Proof.LocalConjugacy.Supersolvable.{u_1} G inst) {p : Nat} [hp : Fact (Nat.Prime p)]
  (hprimes :
    @LocalConjugacy.Proof.LocalConjugacy.HasPrimes.{u_1}
      (@Set.ofPred.{0} Nat fun (r : Nat) => @LE.le.{0} Nat instLENat r p) G inst)
  (P : @Sylow.{u_1} p G inst), @Subgroup.Normal.{u_1} G inst (@Sylow.toSubgroup.{u_1} p G inst 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 27–48; source SHA-256 29509bd9dc03344a0acef30e2bd052c64781f5f1093f7c226e23a6aa0ad6969d.

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