The Schur-Zassenhaus Theorem

In this file we prove the Schur-Zassenhaus theorem.

Main results

• Subgroup.exists_right_complement'_of_coprime: The Schur-Zassenhaus theorem: If H : Subgroup G is normal and has order coprime to its index, then there exists a subgroup K which is a (right) complement of H.
• Subgroup.exists_left_complement'_of_coprime: The Schur-Zassenhaus theorem: If H : Subgroup G is normal and has order coprime to its index, then there exists a subgroup K which is a (left) complement of H.

def Subgroup.QuotientDiff {G : Type u_1} [Group G] (H : Subgroup G) [IsMulCommutative ↥H] [H.FiniteIndex] : Type u_1

The quotient of the transversals of an abelian normal N by the diff relation.

Equations

• H.QuotientDiff = Quotient { r := fun (α β : H.LeftTransversal) => Subgroup.leftTransversals.diff (MonoidHom.id ↥H) α β = 1, iseqv := ⋯ }

instance Subgroup.instInhabitedQuotientDiff {G : Type u_1} [Group G] (H : Subgroup G) [IsMulCommutative ↥H] [H.FiniteIndex] : Inhabited H.QuotientDiff

Equations

• H.instInhabitedQuotientDiff = inferInstanceAs (Inhabited (Quotient { r := fun (α β : H.LeftTransversal) => Subgroup.leftTransversals.diff (MonoidHom.id ↥H) α β = 1, iseqv := ⋯ }))

theorem Subgroup.smul_diff_smul' {G : Type u_1} [Group G] (H : Subgroup G) [IsMulCommutative ↥H] [H.FiniteIndex] (α β : H.LeftTransversal) [hH : H.Normal] (g : Gᵐᵒᵖ) : leftTransversals.diff (MonoidHom.id ↥H) (g • α) (g • β) = ⟨(MulOpposite.unop g)⁻¹ * ↑(leftTransversals.diff (MonoidHom.id ↥H) α β) * MulOpposite.unop g, ⋯⟩

noncomputable instance Subgroup.instMulActionQuotientDiff {G : Type u_1} [Group G] {H : Subgroup G} [IsMulCommutative ↥H] [H.FiniteIndex] [H.Normal] : MulAction G H.QuotientDiff

Equations

• Subgroup.instMulActionQuotientDiff = { smul := fun (g : G) => Quotient.map' (fun (α : H.LeftTransversal) => MulOpposite.op g⁻¹ • α) ⋯, one_smul := ⋯, mul_smul := ⋯ }

theorem Subgroup.smul_diff' {G : Type u_1} [Group G] {H : Subgroup G} [IsMulCommutative ↥H] [H.FiniteIndex] (α β : H.LeftTransversal) [H.Normal] (h : ↥H) : leftTransversals.diff (MonoidHom.id ↥H) α (MulOpposite.op ↑h • β) = leftTransversals.diff (MonoidHom.id ↥H) α β * h ^ H.index

theorem Subgroup.eq_one_of_smul_eq_one {G : Type u_1} [Group G] {H : Subgroup G} [IsMulCommutative ↥H] [H.FiniteIndex] [H.Normal] (hH : (Nat.card ↥H).Coprime H.index) (α : H.QuotientDiff) (h : ↥H) : h • α = α → h = 1

theorem Subgroup.exists_smul_eq {G : Type u_1} [Group G] {H : Subgroup G} [IsMulCommutative ↥H] [H.FiniteIndex] [H.Normal] (hH : (Nat.card ↥H).Coprime H.index) (α β : H.QuotientDiff) : ∃ (h : ↥H), h • α = β

theorem Subgroup.isComplement'_stabilizer_of_coprime {G : Type u_1} [Group G] {H : Subgroup G} [IsMulCommutative ↥H] [H.FiniteIndex] [H.Normal] {α : H.QuotientDiff} (hH : (Nat.card ↥H).Coprime H.index) : H.IsComplement' (MulAction.stabilizer G α)

Proof of the Schur-Zassenhaus theorem

In this section, we prove the Schur-Zassenhaus theorem. The proof is by contradiction. We assume that G is a minimal counterexample to the theorem.

We will arrive at a contradiction via the following steps:

• step 0: N (the normal Hall subgroup) is nontrivial.
• step 1: If K is a subgroup of G with K ⊔ N = ⊤, then K = ⊤.
• step 2: N is a minimal normal subgroup, phrased in terms of subgroups of G.
• step 3: N is a minimal normal subgroup, phrased in terms of subgroups of N.
• step 4: p (min_fact (Fintype.card N)) is prime (follows from step0).
• step 5: P (a Sylow p-subgroup of N) is nontrivial.
• step 6: N is a p-group (applies step 1 to the normalizer of P in G).
• step 7: N is abelian (applies step 3 to the center of N).

theorem Subgroup.SchurZassenhausInduction.step7 {G : Type u} [Group G] {N : Subgroup G} [N.Normal] (h1 : (Nat.card ↥N).Coprime N.index) (h2 : ∀ (G' : Type u) [inst : Group G'] [Finite G'], Nat.card G' < Nat.card G → ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ (H' : Subgroup G'), N'.IsComplement' H') (h3 : ∀ (H : Subgroup G), ¬N.IsComplement' H) [Finite G] : IsMulCommutative ↥N

Do not use this lemma: It is made obsolete by exists_right_complement'_of_coprime

theorem Subgroup.exists_right_complement'_of_coprime {G : Type u} [Group G] {N : Subgroup G} [N.Normal] (hN : (Nat.card ↥N).Coprime N.index) : ∃ (H : Subgroup G), N.IsComplement' H

Schur-Zassenhaus for normal subgroups: If H : Subgroup G is normal, and has order coprime to its index, then there exists a subgroup K which is a (right) complement of H.

theorem Subgroup.exists_left_complement'_of_coprime {G : Type u} [Group G] {N : Subgroup G} [N.Normal] (hN : (Nat.card ↥N).Coprime N.index) : ∃ (H : Subgroup G), H.IsComplement' N

Schur-Zassenhaus for normal subgroups: If H : Subgroup G is normal, and has order coprime to its index, then there exists a subgroup K which is a (left) complement of H.