refactor(GroupTheory/SpecificGroups/Cyclic): Switch from Fintype to…

… `Finite` (#19109)

This PR switches most of `GroupTheory/SpecificGroups/Cyclic.lean` from `Fintype` to `Finite`.

Mathlib/FieldTheory/Finite/Basic.lean

@@ -172,6 +172,7 @@ theorem sum_subgroup_units [Ring K] [NoZeroDivisors K]
 theorem sum_subgroup_pow_eq_zero [CommRing K] [NoZeroDivisors K]
     {G : Subgroup Kˣ} [Fintype G] {k : ℕ} (k_pos : k ≠ 0) (k_lt_card_G : k < Fintype.card G) :
     ∑ x : G, ((x : Kˣ) : K) ^ k = 0 := by
+  rw [← Nat.card_eq_fintype_card] at k_lt_card_G
   nontriviality K
   have := NoZeroDivisors.to_isDomain K
   rcases (exists_pow_ne_one_of_isCyclic k_pos k_lt_card_G) with ⟨a, ha⟩
@@ -255,7 +256,7 @@ theorem cast_card_eq_zero : (q : K) = 0 := by
 theorem forall_pow_eq_one_iff (i : ℕ) : (∀ x : Kˣ, x ^ i = 1) ↔ q - 1 ∣ i := by
   classical
     obtain ⟨x, hx⟩ := IsCyclic.exists_generator (α := Kˣ)
-    rw [← Fintype.card_units, ← orderOf_eq_card_of_forall_mem_zpowers hx,
+    rw [← Nat.card_eq_fintype_card, ← Nat.card_units, ← orderOf_eq_card_of_forall_mem_zpowers hx,
       orderOf_dvd_iff_pow_eq_one]
     constructor
     · intro h; apply h
@@ -586,11 +587,12 @@ theorem unit_isSquare_iff (hF : ringChar F ≠ 2) (a : Fˣ) :
       push_cast
       exact FiniteField.pow_card_sub_one_eq_one (y : F) (Units.ne_zero y)
     · subst a; intro h
-      have key : 2 * (Fintype.card F / 2) ∣ n * (Fintype.card F / 2) := by
+      rw [← Nat.card_eq_fintype_card] at hodd h
+      have key : 2 * (Nat.card F / 2) ∣ n * (Nat.card F / 2) := by
         rw [← pow_mul] at h
-        rw [hodd, ← Fintype.card_units, ← orderOf_eq_card_of_forall_mem_zpowers hg]
+        rw [hodd, ← Nat.card_units, ← orderOf_eq_card_of_forall_mem_zpowers hg]
         apply orderOf_dvd_of_pow_eq_one h
-      have : 0 < Fintype.card F / 2 := Nat.div_pos Fintype.one_lt_card (by norm_num)
+      have : 0 < Nat.card F / 2 := Nat.div_pos Finite.one_lt_card (by norm_num)
       obtain ⟨m, rfl⟩ := Nat.dvd_of_mul_dvd_mul_right this key
       refine ⟨g ^ m, ?_⟩
       dsimp

Mathlib/GroupTheory/Exponent.lean

@@ -501,9 +501,11 @@ variable [Group G]
 lemma Group.one_lt_exponent [Finite G] [Nontrivial G] : 1 < Monoid.exponent G :=
   Monoid.one_lt_exponent
 
+@[to_additive]
 theorem Group.exponent_dvd_card [Fintype G] : Monoid.exponent G ∣ Fintype.card G :=
   Monoid.exponent_dvd.mpr <| fun _ => orderOf_dvd_card
 
+@[to_additive]
 theorem Group.exponent_dvd_nat_card : Monoid.exponent G ∣ Nat.card G :=
   Monoid.exponent_dvd.mpr orderOf_dvd_natCard
 

Mathlib/GroupTheory/SpecificGroups/Quaternion.lean

@@ -210,7 +210,7 @@ theorem orderOf_xa [NeZero n] (i : ZMod (2 * n)) : orderOf (xa i) = 4 := by
 /-- In the special case `n = 1`, `Quaternion 1` is a cyclic group (of order `4`). -/
 theorem quaternionGroup_one_isCyclic : IsCyclic (QuaternionGroup 1) := by
   apply isCyclic_of_orderOf_eq_card
-  · rw [card, mul_one]
+  · rw [Nat.card_eq_fintype_card, card, mul_one]
     exact orderOf_xa 0
 
 /-- If `0 < n`, then `a 1` has order `2 * n`.

Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean

@@ -119,7 +119,7 @@ theorem toFun_spec'' (g : L ≃+* L) {n : ℕ} [NeZero n] {t : L} (ht : IsPrimit
 /-- If g(t)=t^c for all roots of unity, then c=χ(g). -/
 theorem toFun_unique (g : L ≃+* L) (c : ZMod (Fintype.card (rootsOfUnity n L)))
     (hc : ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ c.val : Lˣ)) : c = χ₀ n g := by
-  apply IsCyclic.ext rfl (fun ζ ↦ ?_)
+  apply IsCyclic.ext Nat.card_eq_fintype_card (fun ζ ↦ ?_)
   specialize hc ζ
   suffices ((ζ ^ c.val : Lˣ) : L) = (ζ ^ (χ₀ n g).val : Lˣ) by exact_mod_cast this
   rw [← toFun_spec g ζ, hc]

Mathlib/NumberTheory/LucasPrimality.lean

@@ -54,7 +54,8 @@ theorem reverse_lucas_primality (p : ℕ) (hP : p.Prime) :
   have : Fact p.Prime := ⟨hP⟩
   obtain ⟨g, hg⟩ := IsCyclic.exists_generator (α := (ZMod p)ˣ)
   have h1 : orderOf g = p - 1 := by
-    rwa [orderOf_eq_card_of_forall_mem_zpowers hg, ← Nat.prime_iff_card_units]
+    rwa [orderOf_eq_card_of_forall_mem_zpowers hg, Nat.card_eq_fintype_card,
+      ← Nat.prime_iff_card_units]
   have h2 := tsub_pos_iff_lt.2 hP.one_lt
   rw [← orderOf_injective (Units.coeHom _) Units.ext _, orderOf_eq_iff h2] at h1
   refine ⟨g, h1.1, fun q hq hqd ↦ ?_⟩

Mathlib/NumberTheory/MulChar/Lemmas.lean

@@ -82,7 +82,7 @@ noncomputable def ofRootOfUnity {ζ : Rˣ} (hζ : ζ ∈ rootsOfUnity (Fintype.c
   have : orderOf ζ ∣ Fintype.card Mˣ :=
     orderOf_dvd_iff_pow_eq_one.mpr <| (mem_rootsOfUnity _ ζ).mp hζ
   refine ofUnitHom <| monoidHomOfForallMemZpowers hg <| this.trans <| dvd_of_eq ?_
-  rw [orderOf_generator_eq_natCard hg, Nat.card_eq_fintype_card]
+  rw [orderOf_eq_card_of_forall_mem_zpowers hg, Nat.card_eq_fintype_card]
 
 lemma ofRootOfUnity_spec {ζ : Rˣ} (hζ : ζ ∈ rootsOfUnity (Fintype.card Mˣ) R)
     {g : Mˣ} (hg : ∀ x, x ∈ Subgroup.zpowers g) :

Mathlib/RingTheory/RootsOfUnity/Basic.lean

@@ -251,12 +251,12 @@ namespace IsCyclic
 `n` into another group `G'` to the group of `n`th roots of unity in `G'` determined by a generator
 `g` of `G`. It sends `φ : G →* G'` to `φ g`. -/
 noncomputable
-def monoidHomMulEquivRootsOfUnityOfGenerator {G : Type*} [CommGroup G] [Fintype G] {g : G}
+def monoidHomMulEquivRootsOfUnityOfGenerator {G : Type*} [CommGroup G] {g : G}
     (hg : ∀ (x : G), x ∈ Subgroup.zpowers g) (G' : Type*) [CommGroup G'] :
-    (G →* G') ≃* rootsOfUnity (Fintype.card G) G' where
+    (G →* G') ≃* rootsOfUnity (Nat.card G) G' where
   toFun φ := ⟨(IsUnit.map φ <| Group.isUnit g).unit, by
     simp only [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, IsUnit.unit_spec,
-      ← map_pow, pow_card_eq_one, map_one, Units.val_one]⟩
+      ← map_pow, pow_card_eq_one', map_one, Units.val_one]⟩
   invFun ζ := monoidHomOfForallMemZpowers hg (g' := (ζ.val : G')) <| by
     simpa only [orderOf_eq_card_of_forall_mem_zpowers hg, orderOf_dvd_iff_pow_eq_one,
       ← Units.val_pow_eq_pow_val, Units.val_eq_one] using ζ.prop
@@ -270,12 +270,11 @@ def monoidHomMulEquivRootsOfUnityOfGenerator {G : Type*} [CommGroup G] [Fintype
 
 /-- The group of group homomorphisms from a finite cyclic group `G` of order `n` into another
 group `G'` is (noncanonically) isomorphic to the group of `n`th roots of unity in `G'`. -/
-lemma monoidHom_mulEquiv_rootsOfUnity (G : Type*) [CommGroup G] [Finite G] [IsCyclic G]
+lemma monoidHom_mulEquiv_rootsOfUnity (G : Type*) [CommGroup G] [IsCyclic G]
     (G' : Type*) [CommGroup G'] :
     Nonempty <| (G →* G') ≃* rootsOfUnity (Nat.card G) G' := by
   obtain ⟨g, hg⟩ := IsCyclic.exists_generator (α := G)
-  have : Fintype G := Fintype.ofFinite _
-  exact ⟨Nat.card_eq_fintype_card (α  := G) ▸ monoidHomMulEquivRootsOfUnityOfGenerator hg G'⟩
+  exact ⟨monoidHomMulEquivRootsOfUnityOfGenerator hg G'⟩
 
 end IsCyclic
 

Mathlib/RingTheory/RootsOfUnity/EnoughRootsOfUnity.lean

@@ -64,8 +64,7 @@ lemma natCard_rootsOfUnity (M : Type*) [CommMonoid M] (n : ℕ) [NeZero n]
     [HasEnoughRootsOfUnity M n] :
     Nat.card (rootsOfUnity n M) = n := by
   obtain ⟨ζ, h⟩ := exists_primitiveRoot M n
-  have : Fintype <| rootsOfUnity n M := Fintype.ofFinite _
-  rw [Nat.card_eq_fintype_card, ← IsCyclic.exponent_eq_card]
+  rw [← IsCyclic.exponent_eq_card]
   refine dvd_antisymm ?_ ?_
   · exact Monoid.exponent_dvd_of_forall_pow_eq_one fun g ↦ OneMemClass.coe_eq_one.mp g.prop
   · nth_rewrite 1 [h.eq_orderOf]