feat(group_theory/noncomm_pi_coprod): homomorphism from pi monoids or…

… groups (#11744)

Co-authored-by: Junyan Xu <junyanxumath@gmail.com>



Co-authored-by: Junyan Xu <junyanxumath@gmail.com>

src/group_theory/noncomm_pi_coprod.lean

@@ -0,0 +1,275 @@
+/-
+Copyright (c) 2022 Joachim Breitner. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joachim Breitner
+-/
+import group_theory.order_of_element
+import data.finset.noncomm_prod
+import data.fintype.card
+
+/-!
+# Canonical homomorphism from a finite family of monoids
+
+This file defines the construction of the canonical homomorphism from a family of monoids.
+
+Given a family of morphisms `ϕ i : N i →* M` for each `i : ι` where elements in the
+images of different morphisms commute, we obtain a canonical morphism
+`monoid_hom.noncomm_pi_coprod : (Π i, N i) →* M` that coincides with `ϕ`
+
+## Main definitions
+
+* `monoid_hom.noncomm_pi_coprod : (Π i, N i) →* M` is the main homomorphism
+* `subgroup.noncomm_pi_coprod : (Π i, H i) →* G` is the specialization to `H i : subgroup G`
+   and the subgroup embedding.
+
+## Main theorems
+
+* `monoid_hom.noncomm_pi_coprod` coincides with `ϕ i` when restricted to `N i`
+* `monoid_hom.noncomm_pi_coprod_mrange`: The range of `monoid_hom.noncomm_pi_coprod` is
+  `⨆ (i : ι), (ϕ i).mrange`
+* `monoid_hom.noncomm_pi_coprod_range`: The range of `monoid_hom.noncomm_pi_coprod` is
+  `⨆ (i : ι), (ϕ i).range`
+* `subgroup.noncomm_pi_coprod_range`: The range of `subgroup.noncomm_pi_coprod` is `⨆ (i : ι), H i`.
+* `monoid_hom.injective_noncomm_pi_coprod_of_independent`: in the case of groups, `pi_hom.hom` is
+   injective if the `ϕ` are injective and the ranges of the `ϕ` are independent.
+* `monoid_hom.independent_range_of_coprime_order`: If the `N i` have coprime orders, then the ranges
+   of the `ϕ` are independent.
+* `subgroup.independent_of_coprime_order`: If commuting normal subgroups `H i` have coprime orders,
+   they are independent.
+
+-/
+
+open_locale big_operators
+
+section family_of_monoids
+
+variables {M : Type*} [monoid M]
+
+-- We have a family of monoids
+-- The fintype assumption is not always used, but declared here, to keep things in order
+variables {ι : Type*} [hdec : decidable_eq ι] [fintype ι]
+variables {N : ι → Type*} [∀ i, monoid (N i)]
+
+-- And morphisms ϕ into G
+variables (ϕ : Π (i : ι), N i →* M)
+
+-- We assume that the elements of different morphism commute
+variables (hcomm : ∀ (i j : ι), i ≠ j → ∀ (x : N i) (y : N j), commute (ϕ i x) (ϕ j y))
+include hcomm
+
+-- We use `f` and `g` to denote elements of `Π (i : ι), N i`
+variables (f g : Π (i : ι), N i)
+
+namespace monoid_hom
+
+/-- The canonical homomorphism from a family of monoids. -/
+@[to_additive "The canonical homomorphism from a family of additive monoids.
+
+See also `linear_map.lsum` for a linear version without the commutativity assumption."]
+def noncomm_pi_coprod : (Π (i : ι), N i) →* M :=
+{ to_fun := λ f, finset.univ.noncomm_prod (λ i, ϕ i (f i)) $
+    by { rintros i - j -, by_cases h : i = j, { subst h }, { exact hcomm _ _ h _ _ } },
+  map_one' := by {apply (finset.noncomm_prod_eq_pow_card _ _ _ _ _).trans (one_pow _), simp},
+  map_mul' := λ f g,
+  begin
+    classical,
+    convert @finset.noncomm_prod_mul_distrib _ _ _ _ (λ i, ϕ i (f i)) (λ i, ϕ i (g i)) _ _ _,
+    { ext i, exact map_mul (ϕ i) (f i) (g i), },
+    { rintros i - j - h, exact hcomm _ _ h _ _ },
+  end }
+
+variable {hcomm}
+
+include hdec
+
+@[simp, to_additive]
+lemma noncomm_pi_coprod_mul_single (i : ι) (y : N i):
+  noncomm_pi_coprod ϕ hcomm (pi.mul_single i y) = ϕ i y :=
+begin
+  change finset.univ.noncomm_prod (λ j, ϕ j (pi.mul_single i y j)) _ = ϕ i y,
+  simp only [←finset.insert_erase (finset.mem_univ i)] {single_pass := tt},
+  rw finset.noncomm_prod_insert_of_not_mem _ _ _ _ (finset.not_mem_erase i _),
+  rw pi.mul_single_eq_same,
+  rw finset.noncomm_prod_eq_pow_card,
+  { rw one_pow, exact mul_one _  },
+  { intros j hj, simp only [finset.mem_erase] at hj, simp [hj], },
+end
+
+omit hcomm
+
+/-- The universal property of `noncomm_pi_coprod` -/
+@[to_additive "The universal property of `noncomm_pi_coprod`"]
+def noncomm_pi_coprod_equiv :
+  {ϕ : Π i, N i →* M // pairwise (λ i j, ∀ x y, commute (ϕ i x) (ϕ j y)) }
+    ≃ ((Π i, N i) →* M) :=
+{ to_fun := λ ϕ, noncomm_pi_coprod ϕ.1 ϕ.2,
+  inv_fun := λ f,
+  ⟨ λ i, f.comp (monoid_hom.single N i),
+    λ i j hij x y, commute.map (pi.mul_single_commute i j hij x y) f ⟩,
+  left_inv := λ ϕ, by { ext, simp, },
+  right_inv := λ f, pi_ext (λ i x, by simp) }
+
+omit hdec
+
+include hcomm
+
+@[to_additive]
+lemma noncomm_pi_coprod_mrange : (noncomm_pi_coprod ϕ hcomm).mrange = ⨆ i : ι, (ϕ i).mrange :=
+begin
+  classical,
+  apply le_antisymm,
+  { rintro x ⟨f, rfl⟩,
+    refine submonoid.noncomm_prod_mem _ _ _ _ _,
+    intros i hi,
+    apply submonoid.mem_Sup_of_mem, { use i },
+    simp, },
+  { refine supr_le _,
+    rintro i x ⟨y, rfl⟩,
+    refine ⟨pi.mul_single i y, noncomm_pi_coprod_mul_single _ _ _⟩, },
+end
+
+end monoid_hom
+
+end family_of_monoids
+
+section family_of_groups
+
+variables {G : Type*} [group G]
+variables {ι : Type*} [hdec : decidable_eq ι] [hfin : fintype ι]
+variables {H : ι → Type*} [∀ i, group (H i)]
+variables (ϕ : Π (i : ι), H i →* G)
+variables {hcomm : ∀ (i j : ι), i ≠ j → ∀ (x : H i) (y : H j), commute (ϕ i x) (ϕ j y)}
+include hcomm
+
+-- We use `f` and `g` to denote elements of `Π (i : ι), H i`
+variables (f g : Π (i : ι), H i)
+
+include hfin
+
+namespace monoid_hom
+
+-- The subgroup version of `noncomm_pi_coprod_mrange`
+@[to_additive]
+lemma noncomm_pi_coprod_range : (noncomm_pi_coprod ϕ hcomm).range = ⨆ i : ι, (ϕ i).range :=
+begin
+  classical,
+  apply le_antisymm,
+  { rintro x ⟨f, rfl⟩,
+    refine subgroup.noncomm_prod_mem _ _ _,
+    intros i hi,
+    apply subgroup.mem_Sup_of_mem, { use i },
+    simp, },
+  { refine supr_le _,
+    rintro i x ⟨y, rfl⟩,
+    refine ⟨pi.mul_single i y, noncomm_pi_coprod_mul_single _ _ _⟩, },
+end
+
+@[to_additive]
+lemma injective_noncomm_pi_coprod_of_independent
+  (hind : complete_lattice.independent (λ i, (ϕ i).range))
+  (hinj : ∀ i, function.injective (ϕ i)) :
+  function.injective (noncomm_pi_coprod ϕ hcomm):=
+begin
+  classical,
+  apply (monoid_hom.ker_eq_bot_iff _).mp,
+  apply eq_bot_iff.mpr,
+  intros f heq1,
+  change finset.univ.noncomm_prod (λ i, ϕ i (f i)) _ = 1 at heq1,
+  change f = 1,
+  have : ∀ i, i ∈ finset.univ → ϕ i (f i) = 1 :=
+    subgroup.eq_one_of_noncomm_prod_eq_one_of_independent _ _ _ _ hind (by simp) heq1,
+  ext i,
+  apply hinj,
+  simp [this i (finset.mem_univ i)],
+end
+
+variable (hcomm)
+
+@[to_additive]
+lemma independent_range_of_coprime_order [∀ i, fintype (H i)]
+  (hcoprime : ∀ i j, i ≠ j → nat.coprime (fintype.card (H i)) (fintype.card (H j))) :
+  complete_lattice.independent (λ i, (ϕ i).range) :=
+begin
+  classical,
+  rintros i f ⟨hxi, hxp⟩, dsimp at hxi hxp,
+  rw [supr_subtype', ← noncomm_pi_coprod_range] at hxp,
+  rotate, { intros _ _ hj, apply hcomm, exact hj ∘ subtype.ext },
+  cases hxp with g hgf, cases hxi with g' hg'f,
+  have hxi : order_of f ∣ fintype.card (H i),
+  { rw ← hg'f, exact (order_of_map_dvd _ _).trans order_of_dvd_card_univ },
+  have hxp : order_of f ∣ ∏ j : {j // j ≠ i}, fintype.card (H j),
+  { rw [← hgf, ← fintype.card_pi], exact (order_of_map_dvd _ _).trans order_of_dvd_card_univ },
+  change f = 1, rw [← pow_one f, ← order_of_dvd_iff_pow_eq_one],
+  convert ← nat.dvd_gcd hxp hxi, rw ← nat.coprime_iff_gcd_eq_one,
+  apply nat.coprime_prod_left, intros j _, apply hcoprime, exact j.2,
+end
+
+end monoid_hom
+
+end family_of_groups
+
+namespace subgroup
+
+-- We have an family of subgroups
+variables {G : Type*} [group G]
+variables {ι : Type*} [hdec : decidable_eq ι] [hfin : fintype ι] {H : ι → subgroup G}
+
+-- Elements of `Π (i : ι), H i` are called `f` and `g` here
+variables (f g : Π (i : ι), H i)
+
+section commuting_subgroups
+
+-- We assume that the elements of different subgroups commute
+variables (hcomm : ∀ (i j : ι), i ≠ j → ∀ (x y : G), x ∈ H i → y ∈ H j → commute x y)
+include hcomm
+
+@[to_additive]
+lemma commute_subtype_of_commute (i j : ι) (hne : i ≠ j) :
+  ∀ (x : H i) (y : H j), commute ((H i).subtype x) ((H j).subtype y) :=
+by { rintros ⟨x, hx⟩ ⟨y, hy⟩, exact hcomm i j hne x y hx hy }
+
+include hfin
+
+/-- The canonical homomorphism from a family of subgroups where elements from different subgroups
+commute -/
+@[to_additive "The canonical homomorphism from a family of additive subgroups where elements from
+different subgroups commute"]
+def noncomm_pi_coprod : (Π (i : ι), H i) →* G :=
+  monoid_hom.noncomm_pi_coprod (λ i, (H i).subtype) (commute_subtype_of_commute hcomm)
+
+variable {hcomm}
+
+include hdec
+
+@[simp, to_additive]
+lemma noncomm_pi_coprod_mul_single (i : ι) (y : H i) :
+  noncomm_pi_coprod hcomm (pi.mul_single i y) = y :=
+by apply monoid_hom.noncomm_pi_coprod_mul_single
+
+omit hdec
+
+@[to_additive]
+lemma noncomm_pi_coprod_range : (noncomm_pi_coprod hcomm).range = ⨆ i : ι, H i :=
+by simp [noncomm_pi_coprod, monoid_hom.noncomm_pi_coprod_range]
+
+@[to_additive]
+lemma injective_noncomm_pi_coprod_of_independent (hind : complete_lattice.independent H) :
+  function.injective (noncomm_pi_coprod hcomm) :=
+begin
+  apply monoid_hom.injective_noncomm_pi_coprod_of_independent,
+  { simpa using hind },
+  { intro i, exact subtype.coe_injective }
+end
+
+@[to_additive]
+lemma independent_of_coprime_order [∀ i, fintype (H i)]
+  (hcoprime : ∀ i j, i ≠ j → nat.coprime (fintype.card (H i)) (fintype.card (H j))) :
+  complete_lattice.independent H :=
+begin
+  simpa using monoid_hom.independent_range_of_coprime_order
+    (λ i, (H i).subtype) (commute_subtype_of_commute hcomm) hcoprime,
+end
+
+end commuting_subgroups
+
+end subgroup

src/group_theory/order_of_element.lean

@@ -152,6 +152,11 @@ is_periodic_pt.minimal_period_dvd ((is_periodic_pt_mul_iff_pow_eq_one _).mpr h)
 lemma order_of_dvd_iff_pow_eq_one {n : ℕ} : order_of x ∣ n ↔ x ^ n = 1 :=
 ⟨λ h, by rw [pow_eq_mod_order_of, nat.mod_eq_zero_of_dvd h, pow_zero], order_of_dvd_of_pow_eq_one⟩
 
+@[to_additive add_order_of_map_dvd]
+lemma order_of_map_dvd {H : Type*} [monoid H] (ψ : G →* H) (x : G) :
+  order_of (ψ x) ∣ order_of x :=
+by { apply order_of_dvd_of_pow_eq_one, rw [←map_pow, pow_order_of_eq_one], apply map_one }
+
 @[to_additive]
 lemma exists_pow_eq_self_of_coprime (h : n.coprime (order_of x)) :
   ∃ m : ℕ, (x ^ n) ^ m = x :=
@@ -333,6 +338,10 @@ begin
       order_of_dvd_iff_pow_eq_one] }
 end
 
+@[simp, to_additive]
+lemma order_of_inv (x : G) : order_of x⁻¹ = order_of x :=
+by simp [order_of_eq_order_of_iff]
+
 @[simp, norm_cast, to_additive] lemma order_of_subgroup {H : subgroup G}
   (y: H) : order_of (y : G) = order_of y :=
 order_of_injective H.subtype subtype.coe_injective y

src/group_theory/subgroup/basic.lean

@@ -2775,9 +2775,9 @@ disjoint_def'.trans ⟨λ h x y hx hy hxy,
 generalizes (one direction of) `subgroup.disjoint_iff_mul_eq_one`. -/
 @[to_additive "`finset.noncomm_sum` is “injective” in `f` if `f` maps into independent subgroups.
 This generalizes (one direction of) `add_subgroup.disjoint_iff_add_eq_zero`. "]
-lemma eq_one_of_noncomm_prod_eq_one_of_independent {β : Type*} [group β]
-  (s : finset G) (f : G → β) (comm : ∀ (x ∈ s) (y ∈ s), commute (f x) (f y))
-  (K : G → subgroup β) (hind : complete_lattice.independent K) (hmem : ∀ (x ∈ s), f x ∈ K x)
+lemma eq_one_of_noncomm_prod_eq_one_of_independent {ι : Type*}
+  (s : finset ι) (f : ι → G) (comm : ∀ (x ∈ s) (y ∈ s), commute (f x) (f y))
+  (K : ι → subgroup G) (hind : complete_lattice.independent K) (hmem : ∀ (x ∈ s), f x ∈ K x)
   (heq1 : s.noncomm_prod f comm = 1) : ∀ (i ∈ s), f i = 1 :=
 begin
   classical,
@@ -2786,14 +2786,14 @@ begin
   { simp, },
   { simp only [finset.forall_mem_insert] at comm hmem,
     specialize ih (λ x hx, (comm.2 x hx).2) hmem.2,
-    have hmem_bsupr: s.noncomm_prod f (λ x hx, (comm.2 x hx).2) ∈ ⨆ (i ∈ (s : set G)), K i,
+    have hmem_bsupr: s.noncomm_prod f (λ x hx, (comm.2 x hx).2) ∈ ⨆ (i ∈ (s : set ι)), K i,
     { refine subgroup.noncomm_prod_mem _ _ _,
       intros x hx,
-      have : K x ≤ ⨆ (i ∈ (s : set G)), K i := le_bsupr x hx,
+      have : K x ≤ ⨆ (i ∈ (s : set ι)), K i := le_bsupr x hx,
       exact this (hmem.2 x hx), },
     intro heq1,
     rw finset.noncomm_prod_insert_of_not_mem _ _ _ _ hnmem at heq1,
-    have hnmem' : i ∉ (s : set G), by simpa,
+    have hnmem' : i ∉ (s : set ι), by simpa,
     obtain ⟨heq1i : f i = 1, heq1S : s.noncomm_prod f _ = 1⟩ :=
       subgroup.disjoint_iff_mul_eq_one.mp (hind.disjoint_bsupr hnmem') hmem.1 hmem_bsupr heq1,
     specialize ih heq1S,