# Documentation

Mathlib.GroupTheory.SchurZassenhaus

# The Schur-Zassenhaus Theorem #

In this file we prove the Schur-Zassenhaus theorem.

## Main results #

• 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.
• 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} [] (H : ) :
Type u_1

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

Instances For
instance Subgroup.instInhabitedQuotientDiff {G : Type u_1} [] (H : ) :
theorem Subgroup.smul_diff_smul' {G : Type u_1} [] (H : ) (α : ) (β : ) [hH : ] (g : Gᵐᵒᵖ) :
Subgroup.leftTransversals.diff (MonoidHom.id { x // x H }) (g α) (g β) = { val := * ↑(Subgroup.leftTransversals.diff (MonoidHom.id { x // x H }) α β) * , property := (_ : * ↑(Subgroup.leftTransversals.diff (MonoidHom.id { x // x H }) α β) * H) }
noncomputable instance Subgroup.instMulActionQuotientDiffToMonoidToDivInvMonoid {G : Type u_1} [] {H : } [] :
theorem Subgroup.smul_diff' {G : Type u_1} [] {H : } (α : ) (β : ) [] (h : { x // x H }) :
theorem Subgroup.eq_one_of_smul_eq_one {G : Type u_1} [] {H : } [] (hH : Nat.Coprime (Nat.card { x // x H }) ()) (α : ) (h : { x // x H }) :
h α = αh = 1
theorem Subgroup.exists_smul_eq {G : Type u_1} [] {H : } [] (hH : Nat.Coprime (Nat.card { x // x H }) ()) (α : ) (β : ) :
h, h α = β
theorem Subgroup.isComplement'_stabilizer_of_coprime {G : Type u_1} [] {H : } [] {α : } (hH : Nat.Coprime (Nat.card { x // x H }) ()) :

## 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} [] [] {N : } [] (h1 : Nat.Coprime (Fintype.card { x // x N }) ()) (h2 : ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'], ∀ {N' : Subgroup G'} [inst_2 : ], Nat.Coprime (Fintype.card { x // x N' }) ()H', ) (h3 : ∀ (H : ), ) :

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

theorem Subgroup.exists_right_complement'_of_coprime_of_fintype {G : Type u} [] [] {N : } [] (hN : Nat.Coprime (Fintype.card { x // x N }) ()) :
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_right_complement'_of_coprime {G : Type u} [] {N : } [] (hN : Nat.Coprime (Nat.card { x // x N }) ()) :
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_of_fintype {G : Type u} [] [] {N : } [] (hN : Nat.Coprime (Fintype.card { x // x N }) ()) :
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 (left) complement of H.

theorem Subgroup.exists_left_complement'_of_coprime {G : Type u} [] {N : } [] (hN : Nat.Coprime (Nat.card { x // x N }) ()) :
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 (left) complement of H.