refactor(group_theory/abelianization): simplify abelianization (#1126)

* feat(group_theory/conjugates) : define conjugates
define group conjugates and normal closure

* feat(algebra/order_functions): generalize strict_mono.monotone (#1022)

* trying to merge

* feat(group_theory\presented_group):  define presented groups

Presented groups are defined as a quotient of a free group by the normal subgroup  the relations generate.

* feat(group_theory\presented_group): define presented groups

Presented groups are defined as a quotient of a free group by the normal subgroup  the relations generate

* Update src/group_theory/presented_group.lean

Co-Authored-By: Keeley Hoek <keeley@hoek.io>

* Uniqueness of extension

* Tidied up to_group.unique

* Removed unnecessary line

* Changed naming

* refactor(group_theory/abelianization): simplify abelianization

The commutator of a group was previously defined using lists.
Now it is defined using `normal_closure`.
This change simplifies some of the proofs

Author: Michael-Howes

src/group_theory/abelianization.lean

@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kenny Lau
+Authors: Kenny Lau, Michael Howes
 
 The functor Grp → Ab which is the left adjoint
 of the forgetful functor Ab → Grp.
@@ -15,25 +15,10 @@ universes u v
 variables (α : Type u) [group α]
 
 def commutator : set α :=
-{ x | ∃ L : list α, (∀ z ∈ L, ∃ p q, p * q * p⁻¹ * q⁻¹ = z) ∧ L.prod = x }
+group.normal_closure {x | ∃ p q, p * q * p⁻¹ * q⁻¹ = x}
 
 instance : normal_subgroup (commutator α) :=
-{ one_mem := ⟨[], by simp⟩,
-  mul_mem := λ x y ⟨L1, HL1, HP1⟩ ⟨L2, HL2, HP2⟩,
-    ⟨L1 ++ L2, list.forall_mem_append.2 ⟨HL1, HL2⟩, by simp*⟩,
-  inv_mem := λ x ⟨L, HL, HP⟩, ⟨L.reverse.map has_inv.inv,
-    λ x hx, let ⟨y, h3, h4⟩ := list.exists_of_mem_map hx in
-      let ⟨p, q, h5⟩ := HL y (list.mem_reverse.1 h3) in
-      ⟨q, p, by rw [← h4, ← h5]; simp [mul_assoc]⟩,
-    by rw ← HP; from list.rec_on L (by simp) (λ hd tl ih,
-      by rw [list.reverse_cons, list.map_append, list.prod_append, ih]; simp)⟩,
-  normal := λ x ⟨L, HL, HP⟩ g, ⟨L.map $ λ z, g * z * g⁻¹,
-    λ x hx, let ⟨y, h3, h4⟩ := list.exists_of_mem_map hx in
-      let ⟨p, q, h5⟩ := HL y h3 in
-      ⟨g * p * g⁻¹, g * q * g⁻¹,
-      by rw [← h4, ← h5]; simp [mul_assoc]⟩,
-    by rw ← HP; from list.rec_on L (by simp) (λ hd tl ih,
-      by rw [list.map_cons, list.prod_cons, ih]; simp [mul_assoc])⟩, }
+group.normal_closure.is_normal
 
 def abelianization : Type u :=
 quotient_group.quotient $ commutator α
@@ -43,11 +28,9 @@ namespace abelianization
 local attribute [instance] quotient_group.left_rel normal_subgroup.to_is_subgroup
 
 instance : comm_group (abelianization α) :=
-{ mul_comm := λ x y, quotient.induction_on₂ x y $ λ m n,
-    quotient.sound $ ⟨[n⁻¹*m⁻¹*n*m],
-      by simp; refine ⟨n⁻¹, m⁻¹, _⟩; simp,
-      by simp [mul_assoc]⟩,
-  .. quotient_group.group _ }
+{ mul_comm := λ x y, quotient.induction_on₂ x y $ λ a b, quotient.sound
+    (group.subset_normal_closure ⟨b⁻¹,a⁻¹, by simp [mul_inv_rev, inv_inv, mul_assoc]⟩),
+.. quotient_group.group _}
 
 variable {α}
 
@@ -61,24 +44,24 @@ section lift
 
 variables {β : Type v} [comm_group β] (f : α → β) [is_group_hom f]
 
+lemma commutator_subset_ker : commutator α ⊆ is_group_hom.ker f  :=
+group.normal_closure_subset (λ x ⟨p,q,w⟩, (is_group_hom.mem_ker f).2
+  (by {rw ←w, simp [is_group_hom.map_mul f, is_group_hom.map_inv f, mul_comm]}))
+
 def lift : abelianization α → β :=
-quotient_group.lift _ f $ λ x ⟨L, HL, hx⟩,
-hx ▸ list.rec_on L (λ _, is_group_hom.map_one f) (λ hd tl HL ih,
-  by rw [list.forall_mem_cons] at ih;
-    rcases ih with ⟨⟨p, q, hpq⟩, ih⟩;
-    specialize HL ih; rw [list.prod_cons, is_group_hom.map_mul f, ← hpq, HL];
-    simp [is_group_hom.map_mul f, is_group_hom.map_inv f, mul_comm]) HL
+quotient_group.lift _ f (λ x h, (is_group_hom.mem_ker f).1 (commutator_subset_ker f h))
 
 instance lift.is_group_hom : is_group_hom (lift f) :=
 quotient_group.is_group_hom_quotient_lift _ _ _
 
-@[simp] lemma lift.of (x : α) : lift f (of x) = f x := rfl
+@[simp] lemma lift.of (x : α) : lift f (of x) = f x :=
+rfl
 
 theorem lift.unique
   (g : abelianization α → β) [is_group_hom g]
   (hg : ∀ x, g (of x) = f x) {x} :
   g x = lift f x :=
-quotient.induction_on x $ λ m, hg m
+quotient_group.induction_on x hg
 
 end lift