The Transfer Homomorphism

In this file we construct the transfer homomorphism.

Main definitions

• diff ϕ S T : The difference of two left transversals S and T under the homomorphism ϕ.
• transfer ϕ : The transfer homomorphism induced by ϕ.
• transferCenterPow: The transfer homomorphism G →* center G.

Main results

• transferCenterPow_apply: The transfer homomorphism G →* center G is given by g ↦ g ^ (center G).index.
• ker_transferSylow_isComplement': Burnside's transfer (or normal p-complement) theorem: If hP : N(P) ≤ C(P), then (transfer P hP).ker is a normal p-complement.

noncomputable def Subgroup.leftTransversals.diff {G : Type u_1} [Group G] {H : Subgroup G} {A : Type u_2} [CommGroup A] (ϕ : ↥H →* A) (S T : H.LeftTransversal) [H.FiniteIndex] : A

The difference of two left transversals

Equations

• Subgroup.leftTransversals.diff ϕ S T = ∏ q : G ⧸ H, ϕ ⟨(↑(⋯.leftQuotientEquiv q))⁻¹ * ↑(⋯.leftQuotientEquiv q), ⋯⟩

noncomputable def AddSubgroup.leftTransversals.diff {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {A : Type u_2} [AddCommGroup A] (ϕ : ↥H →+ A) (S T : H.LeftTransversal) [H.FiniteIndex] : A

The difference of two left transversals

Equations

• AddSubgroup.leftTransversals.diff ϕ S T = ∑ q : G ⧸ H, ϕ ⟨-↑(⋯.leftQuotientEquiv q) + ↑(⋯.leftQuotientEquiv q), ⋯⟩

theorem Subgroup.leftTransversals.diff_mul_diff {G : Type u_1} [Group G] {H : Subgroup G} {A : Type u_2} [CommGroup A] (ϕ : ↥H →* A) (R S T : H.LeftTransversal) [H.FiniteIndex] : diff ϕ R S * diff ϕ S T = diff ϕ R T

theorem AddSubgroup.leftTransversals.diff_add_diff {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {A : Type u_2} [AddCommGroup A] (ϕ : ↥H →+ A) (R S T : H.LeftTransversal) [H.FiniteIndex] : diff ϕ R S + diff ϕ S T = diff ϕ R T

theorem Subgroup.leftTransversals.diff_self {G : Type u_1} [Group G] {H : Subgroup G} {A : Type u_2} [CommGroup A] (ϕ : ↥H →* A) (T : H.LeftTransversal) [H.FiniteIndex] : diff ϕ T T = 1

theorem AddSubgroup.leftTransversals.diff_self {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {A : Type u_2} [AddCommGroup A] (ϕ : ↥H →+ A) (T : H.LeftTransversal) [H.FiniteIndex] : diff ϕ T T = 0

theorem Subgroup.leftTransversals.diff_inv {G : Type u_1} [Group G] {H : Subgroup G} {A : Type u_2} [CommGroup A] (ϕ : ↥H →* A) (S T : H.LeftTransversal) [H.FiniteIndex] : (diff ϕ S T)⁻¹ = diff ϕ T S

theorem AddSubgroup.leftTransversals.diff_neg {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {A : Type u_2} [AddCommGroup A] (ϕ : ↥H →+ A) (S T : H.LeftTransversal) [H.FiniteIndex] : -diff ϕ S T = diff ϕ T S

theorem Subgroup.leftTransversals.smul_diff_smul {G : Type u_1} [Group G] {H : Subgroup G} {A : Type u_2} [CommGroup A] (ϕ : ↥H →* A) (S T : H.LeftTransversal) [H.FiniteIndex] (g : G) : diff ϕ (g • S) (g • T) = diff ϕ S T

theorem AddSubgroup.leftTransversals.vadd_diff_vadd {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {A : Type u_2} [AddCommGroup A] (ϕ : ↥H →+ A) (S T : H.LeftTransversal) [H.FiniteIndex] (g : G) : diff ϕ (g +ᵥ S) (g +ᵥ T) = diff ϕ S T

noncomputable def Subgroup.transferFunction {G : Type u_1} [Group G] (H : Subgroup G) (g : G) : G ⧸ H → G

The transfer transversal as a function. Given a ⟨g⟩-orbit q₀, g • q₀, ..., g ^ (m - 1) • q₀ in G ⧸ H, an element g ^ k • q₀ is mapped to g ^ k • g₀ for a fixed choice of representative g₀ of q₀.

Equations

• H.transferFunction g q = g ^ ((H.quotientEquivSigmaZMod g) q).snd.cast * Quotient.out (Quotient.out ((H.quotientEquivSigmaZMod g) q).fst)

theorem Subgroup.transferFunction_apply {G : Type u_1} [Group G] {H : Subgroup G} (g : G) (q : G ⧸ H) : H.transferFunction g q = g ^ ((H.quotientEquivSigmaZMod g) q).snd.cast * Quotient.out (Quotient.out ((H.quotientEquivSigmaZMod g) q).fst)

theorem Subgroup.coe_transferFunction {G : Type u_1} [Group G] {H : Subgroup G} (g : G) (q : G ⧸ H) : ↑(H.transferFunction g q) = q

def Subgroup.transferSet {G : Type u_1} [Group G] (H : Subgroup G) (g : G) : Set G

The transfer transversal as a set. Contains elements of the form g ^ k • g₀ for fixed choices of representatives g₀ of fixed choices of representatives q₀ of ⟨g⟩-orbits in G ⧸ H.

Equations

• H.transferSet g = Set.range (H.transferFunction g)

theorem Subgroup.mem_transferSet {G : Type u_1} [Group G] {H : Subgroup G} (g : G) (q : G ⧸ H) : H.transferFunction g q ∈ H.transferSet g

def Subgroup.transferTransversal {G : Type u_1} [Group G] (H : Subgroup G) (g : G) : H.LeftTransversal

The transfer transversal. Contains elements of the form g ^ k • g₀ for fixed choices of representatives g₀ of fixed choices of representatives q₀ of ⟨g⟩-orbits in G ⧸ H.

Equations

• H.transferTransversal g = ⟨H.transferSet g, ⋯⟩

theorem Subgroup.transferTransversal_apply {G : Type u_1} [Group G] {H : Subgroup G} (g : G) (q : G ⧸ H) : ↑(⋯.leftQuotientEquiv q) = H.transferFunction g q

theorem Subgroup.transferTransversal_apply' {G : Type u_1} [Group G] {H : Subgroup G} (g : G) (q : MulAction.orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)) (k : ZMod (Function.minimalPeriod (fun (x : G ⧸ H) => g • x) (Quotient.out q))) : ↑(⋯.leftQuotientEquiv (g ^ k.cast • Quotient.out q)) = g ^ k.cast * Quotient.out (Quotient.out q)

theorem Subgroup.transferTransversal_apply'' {G : Type u_1} [Group G] {H : Subgroup G} (g : G) (q : MulAction.orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)) (k : ZMod (Function.minimalPeriod (fun (x : G ⧸ H) => g • x) (Quotient.out q))) : ↑(⋯.leftQuotientEquiv (g ^ k.cast • Quotient.out q)) = if k = 0 then g ^ Function.minimalPeriod (fun (x : G ⧸ H) => g • x) (Quotient.out q) * Quotient.out (Quotient.out q) else g ^ k.cast * Quotient.out (Quotient.out q)

noncomputable def MonoidHom.transfer {G : Type u_1} [Group G] {H : Subgroup G} {A : Type u_2} [CommGroup A] (ϕ : ↥H →* A) [H.FiniteIndex] : G →* A

Given ϕ : H →* A from H : Subgroup G to a commutative group A, the transfer homomorphism is transfer ϕ : G →* A.

Equations

• ϕ.transfer = { toFun := fun (g : G) => Subgroup.leftTransversals.diff ϕ default (g • default), map_one' := ⋯, map_mul' := ⋯ }

noncomputable def AddMonoidHom.transfer {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {A : Type u_2} [AddCommGroup A] (ϕ : ↥H →+ A) [H.FiniteIndex] : G →+ A

Given ϕ : H →+ A from H : AddSubgroup G to an additive commutative group A, the transfer homomorphism is transfer ϕ : G →+ A.

Equations

• ϕ.transfer = { toFun := fun (g : G) => AddSubgroup.leftTransversals.diff ϕ default (g +ᵥ default), map_zero' := ⋯, map_add' := ⋯ }

theorem MonoidHom.transfer_def {G : Type u_1} [Group G] {H : Subgroup G} {A : Type u_2} [CommGroup A] (ϕ : ↥H →* A) (T : H.LeftTransversal) [H.FiniteIndex] (g : G) : ϕ.transfer g = Subgroup.leftTransversals.diff ϕ T (g • T)

theorem AddMonoidHom.transfer_def {G : Type u_1} [AddGroup G] {H : AddSubgroup G} {A : Type u_2} [AddCommGroup A] (ϕ : ↥H →+ A) (T : H.LeftTransversal) [H.FiniteIndex] (g : G) : ϕ.transfer g = AddSubgroup.leftTransversals.diff ϕ T (g +ᵥ T)

theorem MonoidHom.transfer_eq_prod_quotient_orbitRel_zpowers_quot {G : Type u_1} [Group G] {H : Subgroup G} {A : Type u_2} [CommGroup A] (ϕ : ↥H →* A) [H.FiniteIndex] (g : G) [Fintype (Quotient (MulAction.orbitRel (↥(Subgroup.zpowers g)) (G ⧸ H)))] : ϕ.transfer g = ∏ q : Quotient (MulAction.orbitRel (↥(Subgroup.zpowers g)) (G ⧸ H)), ϕ ⟨(Quotient.out q.out)⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) => g • x) q.out * Quotient.out q.out, ⋯⟩

Explicit computation of the transfer homomorphism.

theorem MonoidHom.transfer_eq_pow_aux {G : Type u_1} [Group G] {H : Subgroup G} (g : G) (key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k) : g ^ H.index ∈ H

Auxiliary lemma in order to state transfer_eq_pow.

theorem MonoidHom.transfer_eq_pow {G : Type u_1} [Group G] {H : Subgroup G} {A : Type u_2} [CommGroup A] (ϕ : ↥H →* A) [H.FiniteIndex] (g : G) (key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k) : ϕ.transfer g = ϕ ⟨g ^ H.index, ⋯⟩

theorem MonoidHom.transfer_center_eq_pow {G : Type u_1} [Group G] [(Subgroup.center G).FiniteIndex] (g : G) : (id ↥(Subgroup.center G)).transfer g = ⟨g ^ (Subgroup.center G).index, ⋯⟩

noncomputable def MonoidHom.transferCenterPow (G : Type u_1) [Group G] [(Subgroup.center G).FiniteIndex] : G →* ↥(Subgroup.center G)

The transfer homomorphism G →* center G.

Equations

• MonoidHom.transferCenterPow G = { toFun := fun (g : G) => ⟨g ^ (Subgroup.center G).index, ⋯⟩, map_one' := ⋯, map_mul' := ⋯ }

@[simp]

theorem MonoidHom.transferCenterPow_apply {G : Type u_1} [Group G] [(Subgroup.center G).FiniteIndex] (g : G) : ↑((transferCenterPow G) g) = g ^ (Subgroup.center G).index

noncomputable def MonoidHom.transferSylow {G : Type u_1} [Group G] {p : ℕ} (P : Sylow p G) (hP : (↑P).normalizer ≤ Subgroup.centralizer ↑P) [(↑P).FiniteIndex] : G →* ↥↑P

The homomorphism G →* P in Burnside's transfer theorem.

Equations

• MonoidHom.transferSylow P hP = (MonoidHom.id ↥↑P).transfer

theorem MonoidHom.transferSylow_eq_pow_aux {G : Type u_1} [Group G] {p : ℕ} (P : Sylow p G) (hP : (↑P).normalizer ≤ Subgroup.centralizer ↑P) [Fact (Nat.Prime p)] [Finite (Sylow p G)] (g : G) (hg : g ∈ P) (k : ℕ) (g₀ : G) (h : g₀⁻¹ * g ^ k * g₀ ∈ P) : g₀⁻¹ * g ^ k * g₀ = g ^ k

Auxiliary lemma in order to state transferSylow_eq_pow.

theorem MonoidHom.transferSylow_eq_pow {G : Type u_1} [Group G] {p : ℕ} (P : Sylow p G) (hP : (↑P).normalizer ≤ Subgroup.centralizer ↑P) [Fact (Nat.Prime p)] [Finite (Sylow p G)] [(↑P).FiniteIndex] (g : G) (hg : g ∈ P) : (transferSylow P hP) g = ⟨g ^ (↑P).index, ⋯⟩

theorem MonoidHom.transferSylow_restrict_eq_pow {G : Type u_1} [Group G] {p : ℕ} (P : Sylow p G) (hP : (↑P).normalizer ≤ Subgroup.centralizer ↑P) [Fact (Nat.Prime p)] [Finite (Sylow p G)] [(↑P).FiniteIndex] : ⇑((transferSylow P hP).restrict ↑P) = fun (x : ↥P) => x ^ (↑P).index

theorem MonoidHom.ker_transferSylow_isComplement' {G : Type u_1} [Group G] {p : ℕ} (P : Sylow p G) (hP : (↑P).normalizer ≤ Subgroup.centralizer ↑P) [Fact (Nat.Prime p)] [Finite (Sylow p G)] [(↑P).FiniteIndex] : (transferSylow P hP).ker.IsComplement' ↑P

Burnside's normal p-complement theorem: If N(P) ≤ C(P), then P has a normal complement.

theorem MonoidHom.not_dvd_card_ker_transferSylow {G : Type u_1} [Group G] {p : ℕ} (P : Sylow p G) (hP : (↑P).normalizer ≤ Subgroup.centralizer ↑P) [Fact (Nat.Prime p)] [Finite (Sylow p G)] [(↑P).FiniteIndex] : ¬p ∣ Nat.card ↥(transferSylow P hP).ker

theorem MonoidHom.ker_transferSylow_disjoint {G : Type u_1} [Group G] {p : ℕ} (P : Sylow p G) (hP : (↑P).normalizer ≤ Subgroup.centralizer ↑P) [Fact (Nat.Prime p)] [Finite (Sylow p G)] [(↑P).FiniteIndex] (Q : Subgroup G) (hQ : IsPGroup p ↥Q) : Disjoint (transferSylow P hP).ker Q

theorem IsCyclic.normalizer_le_centralizer {G : Type u_3} [Group G] [Finite G] {p : ℕ} (hp : (Nat.card G).minFac = p) {P : Sylow p G} (hP : IsCyclic ↥↑P) : (↑P).normalizer ≤ Subgroup.centralizer ↑P

theorem IsCyclic.isComplement' {G : Type u_3} [Group G] [Finite G] {p : ℕ} (hp : (Nat.card G).minFac = p) {P : Sylow p G} (hP : IsCyclic ↥↑P) : (MonoidHom.transferSylow P ⋯).ker.IsComplement' ↑P

A cyclic Sylow subgroup for the smallest prime has a normal complement.