group_theory.sylow | Mathlib porting status

(last sync)

fix(group_theory/subgroup/basic): generalize centralizer from subgroup G to set G (#18965)

This is consistent with all the other sub<foo>.centralizer definitions.

This generalization reveals that a lot of downstream results are rather strangely stated about zpowers. This does not attempt to change these, instead leaving the work for a follow up (either in a later mathlib3 PR or in mathlib4).

Diff

@@ -302,18 +302,19 @@ ext (λ g, sylow.smul_eq_iff_mem_normalizer)
 
 lemma sylow.conj_eq_normalizer_conj_of_mem_centralizer
   [fact p.prime] [finite (sylow p G)] (P : sylow p G) (x g : G)
-  (hx : x ∈ (P : subgroup G).centralizer) (hy : g⁻¹ * x * g ∈ (P : subgroup G).centralizer) :
+  (hx : x ∈ centralizer (P : set G)) (hy : g⁻¹ * x * g ∈ centralizer (P : set G)) :
   ∃ n ∈ (P : subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
 begin
-  have h1 : ↑P ≤ (zpowers x).centralizer,
+  have h1 : ↑P ≤ centralizer (zpowers x : set G),
   { rwa [le_centralizer_iff, zpowers_le] },
-  have h2 : ↑(g • P) ≤ (zpowers x).centralizer,
+  have h2 : ↑(g • P) ≤ centralizer (zpowers x : set G),
   { rw [le_centralizer_iff, zpowers_le],
     rintros - ⟨z, hz, rfl⟩,
     specialize hy z hz,
     rwa [←mul_assoc, ←eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ←mul_assoc,
       eq_inv_mul_iff_mul_eq, ←mul_assoc, ←mul_assoc] at hy },
-  obtain ⟨h, hh⟩ := exists_smul_eq (zpowers x).centralizer ((g • P).subtype h2) (P.subtype h1),
+  obtain ⟨h, hh⟩ :=
+    exists_smul_eq (centralizer (zpowers x : set G)) ((g • P).subtype h2) (P.subtype h1),
   simp_rw [sylow.smul_subtype, smul_def, smul_smul] at hh,
   refine ⟨h * g, sylow.smul_eq_iff_mem_normalizer.mp (sylow.subtype_injective hh), _⟩,
   rw [←mul_assoc, commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
@@ -322,7 +323,8 @@ end
 lemma sylow.conj_eq_normalizer_conj_of_mem [fact p.prime] [finite (sylow p G)] (P : sylow p G)
   [hP : (P : subgroup G).is_commutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
   ∃ n ∈ (P : subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
-P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
+P.conj_eq_normalizer_conj_of_mem_centralizer x g
+  (le_centralizer (P : subgroup G) hx : _) (le_centralizer (P : subgroup G) hy : _)
 
 /-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
 noncomputable def sylow.equiv_quotient_normalizer [fact p.prime] [fintype (sylow p G)]

feat(group_theory/sylow): add inverse to card_eq_multiplicity (#18300)

The lemma card_eq_multiplicity states that a Sylow group of a finite group has cardinality p^n, where n is the multiplicity of p in the group order. This PR adds an inverse definition card_eq_multiplicity_to_sylow, promoting a subgroup of the right cardinality to a Sylow group, and a simplification lemma coe_card_eq_multiplicity_to_sylow for the coercion of the resulting Sylow group back to a subgroup.

Diff

@@ -36,7 +36,7 @@ The Sylow theorems are the following results for every finite group `G` and ever
   there exists a subgroup of `G` of order `pⁿ`.
 * `is_p_group.exists_le_sylow`: A generalization of Sylow's first theorem:
   Every `p`-subgroup is contained in a Sylow `p`-subgroup.
-* `sylow.card_eq_multiplicity`: The cardinality of a Sylow group is `p ^ n`
+* `sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
  where `n` is the multiplicity of `p` in the group order.
 * `sylow_conjugate`: A generalization of Sylow's second theorem:
   If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
@@ -592,7 +592,7 @@ begin
   rwa [h, card_bot] at key,
 end
 
-/-- The cardinality of a Sylow group is `p ^ n`
+/-- The cardinality of a Sylow subgroup is `p ^ n`
  where `n` is the multiplicity of `p` in the group order. -/
 lemma card_eq_multiplicity [fintype G] {p : ℕ} [hp : fact p.prime] (P : sylow p G) :
   card P = p ^ nat.factorization (card G) p :=
@@ -603,6 +603,21 @@ begin
   exact P.1.card_subgroup_dvd_card,
 end
 
+/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
+ where `n` is the multiplicity of `p` in the group order. -/
+def of_card [fintype G] {p : ℕ} [hp : fact p.prime] (H : subgroup G) [fintype H]
+  (card_eq : card H = p ^ (card G).factorization p) : sylow p G :=
+{ to_subgroup := H,
+  is_p_group' := is_p_group.of_card card_eq,
+  is_maximal' := begin
+    obtain ⟨P, hHP⟩ := (is_p_group.of_card card_eq).exists_le_sylow,
+    exact set_like.ext' (set.eq_of_subset_of_card_le hHP
+      (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ λ _, P.3,
+  end }
+
+@[simp, norm_cast] lemma coe_of_card [fintype G] {p : ℕ} [hp : fact p.prime] (H : subgroup G)
+  [fintype H] (card_eq : card H = p ^ (card G).factorization p) : ↑(of_card H card_eq) = H := rfl
+
 lemma subsingleton_of_normal {p : ℕ} [fact p.prime] [finite (sylow p G)] (P : sylow p G)
   (h : (P : subgroup G).normal) : subsingleton (sylow p G) :=
 begin
@@ -667,8 +682,8 @@ normalizer_eq_top.mp $ normalizer_condition_iff_only_full_group_self_normalizing
 
 open_locale big_operators
 
-/-- If all its sylow groups are normal, then a finite group is isomorphic to the direct product
-of these sylow groups.
+/-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
+of these Sylow subgroups.
 -/
 noncomputable
 def direct_product_of_normal [fintype G]
@@ -677,7 +692,7 @@ def direct_product_of_normal [fintype G]
 begin
   set ps := (fintype.card G).factorization.support,
 
-  -- “The” sylow group for p
+  -- “The” Sylow subgroup for p
   let P : Π p, sylow p G := default,
 
   have hcomm : pairwise (λ (p₁ p₂ : ps), ∀ (x y : G), x ∈ P p₁ → y ∈ P p₂ → commute x y),
@@ -689,7 +704,7 @@ begin
     apply is_p_group.disjoint_of_ne p₁ p₂ hne' _ _ (P p₁).is_p_group' (P p₂).is_p_group', },
 
   refine mul_equiv.trans _ _,
-  -- There is only one sylow group for each p, so the inner product is trivial
+  -- There is only one Sylow subgroup for each p, so the inner product is trivial
   show (Π p : ps, Π P : sylow p G, P) ≃* (Π p : ps, P p),
   { -- here we need to help the elaborator with an explicit instantiation
     apply @mul_equiv.Pi_congr_right ps (λ p, (Π P : sylow p G, P)) (λ p, P p) _ _ ,

(first ported)

Changes in mathlib3port

mathlib3

mathlib3port

bump to nightly-2024-04-06-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

e34029c9

Diff

@@ -660,7 +660,7 @@ theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.
   have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
   rw [card_eq_card_quotient_mul_card_subgroup H,
     card_eq_card_quotient_mul_card_subgroup (H.subgroup_of (normalizer H)), Fintype.card_congr this,
-    hH, pow_succ]
+    hH, pow_succ']
   exact (card_quotient_normalizer_modeq_card_quotient hH).mul_right' _
 #align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
 -/
@@ -675,7 +675,7 @@ theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp :
   have hcard : card (G ⧸ H) = s * p :=
     (mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
       (by
-        rwa [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
+        rwa [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ, mul_assoc, mul_comm p])
   have hm :
     s * p % p =
       card (normalizer H ⧸ Subgroup.comap ((normalizer H).Subtype : normalizer H →* G) H) % p :=
@@ -705,7 +705,7 @@ theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact
   have hcard : card (G ⧸ H) = s * p :=
     (mul_left_inj' (show card H ≠ 0 from Fintype.card_ne_zero)).1
       (by
-        rwa [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p])
+        rwa [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ, mul_assoc, mul_comm p])
   have hm : s * p % p = card (normalizer H ⧸ H.subgroupOf H.normalizer) % p :=
     card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
       hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
@@ -730,7 +730,7 @@ theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact
     rw [Set.card_image_of_injective
         (Subgroup.comap (mk' (H.subgroup_of H.normalizer)) (zpowers x) : Set H.normalizer)
         Subtype.val_injective,
-      pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, Fintype.card_zpowers, ← Fintype.card_prod]
+      pow_succ, ← hH, Fintype.card_congr hequiv, ← hx, Fintype.card_zpowers, ← Fintype.card_prod]
     exact @Fintype.card_congr _ _ (id _) (id _) (preimage_mk_equiv_subgroup_times_set _ _),
     by
     intro y hy

bump to nightly-2024-03-17-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

22f01ce4

Diff

@@ -458,10 +458,10 @@ theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite
     rintro - ⟨z, hz, rfl⟩
     specialize hy z hz
     rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
-      eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy 
+      eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
   obtain ⟨h, hh⟩ :=
     exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).Subtype h2) (P.subtype h1)
-  simp_rw [Sylow.smul_subtype, smul_def, smul_smul] at hh 
+  simp_rw [Sylow.smul_subtype, smul_def, smul_smul] at hh
   refine' ⟨h * g, sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
   rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
 #align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
@@ -520,7 +520,7 @@ theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup
   by
   intro h
   haveI := (P : Subgroup G).fintypeQuotientOfFiniteIndex
-  rw [index_eq_card] at h 
+  rw [index_eq_card] at h
   obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card p h
   have h := IsPGroup.of_card ((order_eq_card_zpowers.symm.trans hx).trans (pow_one p).symm)
   let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
@@ -531,7 +531,7 @@ theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup
   replace hp := mt order_of_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
   rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
     comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
-    QuotientGroup.ker_mk'] at hp 
+    QuotientGroup.ker_mk'] at hp
   exact hp.ne' (P.3 hQ hp.le)
 #align not_dvd_index_sylow' not_dvd_index_sylow'
 -/
@@ -561,7 +561,7 @@ theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N
   rw [← inv_mul_cancel_left (↑n) g, sup_comm]
   apply mul_mem_sup (N.inv_mem n.2)
   rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← mul_aut.conj_normal.map_mul,
-    Sylow.ext_iff, Sylow.pointwise_smul_def, pointwise_smul_def] at hn 
+    Sylow.ext_iff, Sylow.pointwise_smul_def, pointwise_smul_def] at hn
   refine' fun x =>
     (mem_map_iff_mem
             (show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
@@ -613,7 +613,7 @@ theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Fin
     inv_mem_iff.1
       (mem_normalizer_fintype fun n (hn : n ∈ H) =>
         have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha (mem_orbit _ ⟨n⁻¹, H.inv_mem hn⟩))
-        show _ ∈ H by rw [mul_inv_rev, inv_inv] at this ; convert this; rw [inv_inv]),
+        show _ ∈ H by rw [mul_inv_rev, inv_inv] at this; convert this; rw [inv_inv]),
     fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
     (mem_fixedPoints' _).2 fun y =>
       Quotient.inductionOn' y fun y hy =>
@@ -623,7 +623,7 @@ theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Fin
           inv_mem_iff.1 <|
             (hx _).2 <|
               (mul_mem_cancel_left (inv_mem hb₁)).1 <| by
-                rw [hx] at hb₂  <;> simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
+                rw [hx] at hb₂ <;> simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
 #align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
 -/
 
@@ -680,7 +680,7 @@ theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp :
     s * p % p =
       card (normalizer H ⧸ Subgroup.comap ((normalizer H).Subtype : normalizer H →* G) H) % p :=
     hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
-  Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm )
+  Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
 #align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
 -/
 
@@ -710,7 +710,7 @@ theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact
     card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
       hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
   have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
-    Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm )
+    Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
   let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
   have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
   ⟨Subgroup.map (normalizer H).Subtype
@@ -787,9 +787,9 @@ theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime]
 theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
     (hdvd : p ∣ card G) : p ∣ card P :=
   by
-  rw [← pow_one p] at hdvd 
+  rw [← pow_one p] at hdvd
   have key := P.pow_dvd_card_of_pow_dvd_card hdvd
-  rwa [pow_one] at key 
+  rwa [pow_one] at key
 #align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
 -/
 
@@ -808,7 +808,7 @@ theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow
   by
   refine' fun h => hp.out.not_dvd_one _
   have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
-  rwa [h, card_bot] at key 
+  rwa [h, card_bot] at key
 #align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
 -/
 
@@ -858,7 +858,7 @@ theorem unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylo
   intro Q R
   obtain ⟨x, h1⟩ := exists_smul_eq G P Q
   obtain ⟨x, h2⟩ := exists_smul_eq G P R
-  rw [Sylow.smul_eq_of_normal] at h1 h2 
+  rw [Sylow.smul_eq_of_normal] at h1 h2
   rw [← h1, ← h2]
 #align sylow.subsingleton_of_normal Sylow.unique_of_normal
 -/
@@ -895,7 +895,7 @@ theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P :
   by
   have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
   simp_rw [← normalizer_eq_top, coeSubtype, ← subgroup_of_normalizer_eq le_normalizer, ←
-    subgroup_of_normalizer_eq le_rfl, subgroup_of_self] at this 
+    subgroup_of_normalizer_eq le_rfl, subgroup_of_self] at this
   rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map, ← this rfl]
   exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))
 #align sylow.normalizer_normalizer Sylow.normalizer_normalizer

bump to nightly-2024-02-23-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

ac9421d6

Diff

@@ -469,7 +469,7 @@ theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite
 
 #print Sylow.conj_eq_normalizer_conj_of_mem /-
 theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
-    [hP : (P : Subgroup G).IsCommutativeₓ] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
+    [hP : (P : Subgroup G).Commutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
     ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
   P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer (P : Subgroup G) hx : _)
     (le_centralizer (P : Subgroup G) hy : _)

bump to nightly-2024-02-05-08

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

516c6ec2

Diff

@@ -469,7 +469,7 @@ theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite
 
 #print Sylow.conj_eq_normalizer_conj_of_mem /-
 theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
-    [hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
+    [hP : (P : Subgroup G).IsCommutativeₓ] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
     ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
   P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer (P : Subgroup G) hx : _)
     (le_centralizer (P : Subgroup G) hy : _)

bump to nightly-2024-02-01-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

5433bded

Diff

@@ -361,7 +361,27 @@ theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P)
 /-- A generalization of **Sylow's second theorem**.
   If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
 instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
-  ⟨fun P Q => by classical⟩
+  ⟨fun P Q => by
+    classical
+    cases nonempty_fintype (Sylow p G)
+    have H := fun {R : Sylow p G} {S : orbit G P} =>
+      calc
+        S ∈ fixed_points R (orbit G P) ↔ S.1 ∈ fixed_points R (Sylow p G) :=
+          forall_congr' fun a => Subtype.ext_iff
+        _ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
+        _ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
+    suffices Set.Nonempty (fixed_points Q (orbit G P)) by
+      exact Exists.elim this fun R hR => (congr_arg _ (Sylow.ext (H.mp hR))).mp R.2
+    apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
+    refine' fun h => hp.out.not_dvd_one (nat.modeq_zero_iff_dvd.mp _)
+    calc
+      1 = card (fixed_points P (orbit G P)) := _
+      _ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
+      _ ≡ 0 [MOD p] := nat.modeq_zero_iff_dvd.mpr h
+    rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
+    refine' card_congr' (congr_arg _ (Eq.symm _))
+    rw [Set.eq_singleton_iff_unique_mem]
+    exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
 
 variable (p) (G)

bump to nightly-2024-01-31-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

61100fe9

Diff

@@ -361,27 +361,7 @@ theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P)
 /-- A generalization of **Sylow's second theorem**.
   If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
 instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
-  ⟨fun P Q => by
-    classical
-    cases nonempty_fintype (Sylow p G)
-    have H := fun {R : Sylow p G} {S : orbit G P} =>
-      calc
-        S ∈ fixed_points R (orbit G P) ↔ S.1 ∈ fixed_points R (Sylow p G) :=
-          forall_congr' fun a => Subtype.ext_iff
-        _ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
-        _ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
-    suffices Set.Nonempty (fixed_points Q (orbit G P)) by
-      exact Exists.elim this fun R hR => (congr_arg _ (Sylow.ext (H.mp hR))).mp R.2
-    apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
-    refine' fun h => hp.out.not_dvd_one (nat.modeq_zero_iff_dvd.mp _)
-    calc
-      1 = card (fixed_points P (orbit G P)) := _
-      _ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
-      _ ≡ 0 [MOD p] := nat.modeq_zero_iff_dvd.mpr h
-    rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
-    refine' card_congr' (congr_arg _ (Eq.symm _))
-    rw [Set.eq_singleton_iff_unique_mem]
-    exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
+  ⟨fun P Q => by classical⟩
 
 variable (p) (G)

bump to nightly-2023-11-29-11

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

049e2cad

Diff

@@ -730,8 +730,7 @@ theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact
     rw [Set.card_image_of_injective
         (Subgroup.comap (mk' (H.subgroup_of H.normalizer)) (zpowers x) : Set H.normalizer)
         Subtype.val_injective,
-      pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, orderOf_eq_card_zpowers, ←
-      Fintype.card_prod]
+      pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, Fintype.card_zpowers, ← Fintype.card_prod]
     exact @Fintype.card_congr _ _ (id _) (id _) (preimage_mk_equiv_subgroup_times_set _ _),
     by
     intro y hy

bump to nightly-2023-11-15-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

38c8ceef

Diff

@@ -851,8 +851,8 @@ theorem coe_ofCard [Fintype G] {p : ℕ} [hp : Fact p.Prime] (H : Subgroup G) [F
 #align sylow.coe_of_card Sylow.coe_ofCard
 -/
 
-#print Sylow.subsingleton_of_normal /-
-theorem subsingleton_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
+#print Sylow.unique_of_normal /-
+theorem unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
     (h : (P : Subgroup G).Normal) : Subsingleton (Sylow p G) :=
   by
   apply Subsingleton.intro
@@ -861,7 +861,7 @@ theorem subsingleton_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P
   obtain ⟨x, h2⟩ := exists_smul_eq G P R
   rw [Sylow.smul_eq_of_normal] at h1 h2 
   rw [← h1, ← h2]
-#align sylow.subsingleton_of_normal Sylow.subsingleton_of_normal
+#align sylow.subsingleton_of_normal Sylow.unique_of_normal
 -/
 
 section Pointwise
@@ -872,7 +872,7 @@ open scoped Pointwise
 theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
     (h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic :=
   by
-  haveI := Sylow.subsingleton_of_normal P h
+  haveI := Sylow.unique_of_normal P h
   rw [characteristic_iff_map_eq]
   intro Φ
   show (Φ • P).toSubgroup = P.to_subgroup
@@ -941,8 +941,8 @@ noncomputable def directProductOfNormal [Fintype G]
   have hcomm : Pairwise fun p₁ p₂ : ps => ∀ x y : G, x ∈ P p₁ → y ∈ P p₂ → Commute x y :=
     by
     rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne
-    haveI hp₁' := Fact.mk (Nat.prime_of_mem_factorization hp₁)
-    haveI hp₂' := Fact.mk (Nat.prime_of_mem_factorization hp₂)
+    haveI hp₁' := Fact.mk (Nat.prime_of_mem_primeFactors hp₁)
+    haveI hp₂' := Fact.mk (Nat.prime_of_mem_primeFactors hp₂)
     have hne' : p₁ ≠ p₂ := by simpa using hne
     apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P p₁)) (hn (P p₂))
     apply IsPGroup.disjoint_of_ne p₁ p₂ hne' _ _ (P p₁).is_p_group' (P p₂).is_p_group'
@@ -952,10 +952,10 @@ noncomputable def directProductOfNormal [Fintype G]
   · -- here we need to help the elaborator with an explicit instantiation
     apply @MulEquiv.piCongrRight ps (fun p => ∀ P : Sylow p G, P) (fun p => P p) _ _
     rintro ⟨p, hp⟩
-    haveI hp' := Fact.mk (Nat.prime_of_mem_factorization hp)
+    haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
     haveI := subsingleton_of_normal _ (hn (P p))
     change (∀ P : Sylow p G, P) ≃* P p
-    exact MulEquiv.piSubsingleton _ _
+    exact MulEquiv.piUnique _ _
   show (∀ p : ps, P p) ≃* G
   apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
   apply (bijective_iff_injective_and_card _).mpr
@@ -964,8 +964,8 @@ noncomputable def directProductOfNormal [Fintype G]
   · apply Subgroup.injective_noncommPiCoprod_of_independent
     apply independent_of_coprime_order hcomm
     rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne
-    haveI hp₁' := Fact.mk (Nat.prime_of_mem_factorization hp₁)
-    haveI hp₂' := Fact.mk (Nat.prime_of_mem_factorization hp₂)
+    haveI hp₁' := Fact.mk (Nat.prime_of_mem_primeFactors hp₁)
+    haveI hp₂' := Fact.mk (Nat.prime_of_mem_primeFactors hp₂)
     have hne' : p₁ ≠ p₂ := by simpa using hne
     apply IsPGroup.coprime_card_of_ne p₁ p₂ hne' _ _ (P p₁).is_p_group' (P p₂).is_p_group'
   show card (∀ p : ps, P p) = card G
@@ -975,7 +975,7 @@ noncomputable def directProductOfNormal [Fintype G]
       _ = ∏ p : ps, p.1 ^ (card G).factorization p.1 :=
         by
         congr 1 with ⟨p, hp⟩
-        exact @card_eq_multiplicity _ _ _ p ⟨Nat.prime_of_mem_factorization hp⟩ (P p)
+        exact @card_eq_multiplicity _ _ _ p ⟨Nat.prime_of_mem_primeFactors hp⟩ (P p)
       _ = ∏ p in ps, p ^ (card G).factorization p :=
         (Finset.prod_finset_coe (fun p => p ^ (card G).factorization p) _)
       _ = (card G).factorization.Prod pow := rfl

bump to nightly-2023-09-24-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

5655435f

Diff

@@ -3,12 +3,12 @@ Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Thomas Browning
 -/
-import Mathbin.Data.Nat.Factorization.Basic
-import Mathbin.Data.SetLike.Fintype
-import Mathbin.GroupTheory.GroupAction.ConjAct
-import Mathbin.GroupTheory.PGroup
-import Mathbin.GroupTheory.NoncommPiCoprod
-import Mathbin.Order.Atoms.Finite
+import Data.Nat.Factorization.Basic
+import Data.SetLike.Fintype
+import GroupTheory.GroupAction.ConjAct
+import GroupTheory.PGroup
+import GroupTheory.NoncommPiCoprod
+import Order.Atoms.Finite
 
 #align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"

bump to nightly-2023-09-20-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

e28e6964

Diff

@@ -797,7 +797,7 @@ theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G
 #print Sylow.card_coprime_index /-
 /-- Sylow subgroups are Hall subgroups. -/
 theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
-    (card P).coprime (index (P : Subgroup G)) :=
+    (card P).Coprime (index (P : Subgroup G)) :=
   let ⟨n, hn⟩ := IsPGroup.iff_card.mp P.2
   hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
 #align sylow.card_coprime_index Sylow.card_coprime_index

bump to nightly-2023-08-17-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

60285dcc

Diff

@@ -90,7 +90,7 @@ instance : SetLike (Sylow p G) G where
 
 instance : SubgroupClass (Sylow p G) G
     where
-  mul_mem s _ _ := s.mul_mem'
+  hMul_mem s _ _ := s.hMul_mem'
   one_mem s := s.one_mem'
   inv_mem s _ := s.inv_mem'
 
@@ -175,9 +175,9 @@ theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q :
         ⟨{  carrier := ⋃ R : c, R
             one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
             inv_mem' := fun g ⟨_, ⟨R, rfl⟩, hg⟩ => ⟨R, ⟨R, rfl⟩, R.1.inv_mem hg⟩
-            mul_mem' := fun g h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
-              (hc2.Total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.mul_mem (T hg) hh⟩) fun T =>
-                ⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
+            hMul_mem' := fun g h ⟨_, ⟨R, rfl⟩, hg⟩ ⟨_, ⟨S, rfl⟩, hh⟩ =>
+              (hc2.Total R.2 S.2).elim (fun T => ⟨S, ⟨S, rfl⟩, S.1.hMul_mem (T hg) hh⟩) fun T =>
+                ⟨R, ⟨R, rfl⟩, R.1.hMul_mem hg (T hh)⟩ },
           fun ⟨g, _, ⟨S, rfl⟩, hg⟩ =>
           by
           refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
@@ -263,7 +263,7 @@ instance Sylow.pointwiseMulAction {α : Type _} [Group α] [MulDistribMulAction
           (congr_arg (· ∈ g⁻¹ • Q) (inv_smul_smul g s)).mp
             (smul_mem_pointwise_smul (g • s) g⁻¹ Q (hS (smul_mem_pointwise_smul s g P hs))))⟩
   one_smul P := Sylow.ext (one_smul α P)
-  mul_smul g h P := Sylow.ext (mul_smul g h P)
+  hMul_smul g h P := Sylow.ext (hMul_smul g h P)
 #align sylow.pointwise_mul_action Sylow.pointwiseMulAction
 -/

bump to nightly-2023-07-25-03

mathlib commit https://github.com/leanprover-community/mathlib/commit/c0c52abb75074ed8b73a948341f50521fbf43b4c

748c9a7c

Diff

@@ -10,7 +10,7 @@ import Mathbin.GroupTheory.PGroup
 import Mathbin.GroupTheory.NoncommPiCoprod
 import Mathbin.Order.Atoms.Finite
 
-#align_import group_theory.sylow from "leanprover-community/mathlib"@"bd365b1a4901dbd878e86cb146c2bd86533df468"
+#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
 
 /-!
 # Sylow theorems
@@ -447,19 +447,20 @@ theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
 
 #print Sylow.conj_eq_normalizer_conj_of_mem_centralizer /-
 theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
-    (P : Sylow p G) (x g : G) (hx : x ∈ (P : Subgroup G).centralizer)
-    (hy : g⁻¹ * x * g ∈ (P : Subgroup G).centralizer) :
+    (P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
+    (hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
     ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
   by
-  have h1 : ↑P ≤ (zpowers x).centralizer := by rwa [le_centralizer_iff, zpowers_le]
-  have h2 : ↑(g • P) ≤ (zpowers x).centralizer :=
+  have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
+  have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) :=
     by
     rw [le_centralizer_iff, zpowers_le]
     rintro - ⟨z, hz, rfl⟩
     specialize hy z hz
     rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
       eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy 
-  obtain ⟨h, hh⟩ := exists_smul_eq (zpowers x).centralizer ((g • P).Subtype h2) (P.subtype h1)
+  obtain ⟨h, hh⟩ :=
+    exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).Subtype h2) (P.subtype h1)
   simp_rw [Sylow.smul_subtype, smul_def, smul_smul] at hh 
   refine' ⟨h * g, sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
   rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
@@ -470,7 +471,8 @@ theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite
 theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
     [hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
     ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
-  P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
+  P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer (P : Subgroup G) hx : _)
+    (le_centralizer (P : Subgroup G) hy : _)
 #align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
 -/

bump to nightly-2023-07-20-09

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

9c56d0dd

Diff

@@ -2,11 +2,6 @@
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Thomas Browning
-
-! This file was ported from Lean 3 source module group_theory.sylow
-! leanprover-community/mathlib commit bd365b1a4901dbd878e86cb146c2bd86533df468
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Data.Nat.Factorization.Basic
 import Mathbin.Data.SetLike.Fintype
@@ -15,6 +10,8 @@ import Mathbin.GroupTheory.PGroup
 import Mathbin.GroupTheory.NoncommPiCoprod
 import Mathbin.Order.Atoms.Finite
 
+#align_import group_theory.sylow from "leanprover-community/mathlib"@"bd365b1a4901dbd878e86cb146c2bd86533df468"
+
 /-!
 # Sylow theorems

bump to nightly-2023-07-16-17

mathlib commit https://github.com/leanprover-community/mathlib/commit/2fe465deb81bcd7ccafa065bb686888a82f15372

6db63fa6

Diff

@@ -827,6 +827,7 @@ theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylo
 #align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
 -/
 
+#print Sylow.ofCard /-
 /-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
  where `n` is the multiplicity of `p` in the group order. -/
 def ofCard [Fintype G] {p : ℕ} [hp : Fact p.Prime] (H : Subgroup G) [Fintype H]
@@ -841,12 +842,15 @@ def ofCard [Fintype G] {p : ℕ} [hp : Fact p.Prime] (H : Subgroup G) [Fintype H
           (Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸
         fun _ => P.3
 #align sylow.of_card Sylow.ofCard
+-/
 
+#print Sylow.coe_ofCard /-
 @[simp, norm_cast]
 theorem coe_ofCard [Fintype G] {p : ℕ} [hp : Fact p.Prime] (H : Subgroup G) [Fintype H]
     (card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
   rfl
 #align sylow.coe_of_card Sylow.coe_ofCard
+-/
 
 #print Sylow.subsingleton_of_normal /-
 theorem subsingleton_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)

bump to nightly-2023-07-12-03

mathlib commit https://github.com/leanprover-community/mathlib/commit/4e24c4bfcff371c71f7ba22050308aa17815626c

403fa41d

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Thomas Browning
 
 ! This file was ported from Lean 3 source module group_theory.sylow
-! leanprover-community/mathlib commit f60c6087a7275b72d5db3c5a1d0e19e35a429c0a
+! leanprover-community/mathlib commit bd365b1a4901dbd878e86cb146c2bd86533df468
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -40,7 +40,7 @@ The Sylow theorems are the following results for every finite group `G` and ever
   there exists a subgroup of `G` of order `pⁿ`.
 * `is_p_group.exists_le_sylow`: A generalization of Sylow's first theorem:
   Every `p`-subgroup is contained in a Sylow `p`-subgroup.
-* `sylow.card_eq_multiplicity`: The cardinality of a Sylow group is `p ^ n`
+* `sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
  where `n` is the multiplicity of `p` in the group order.
 * `sylow_conjugate`: A generalization of Sylow's second theorem:
   If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
@@ -815,7 +815,7 @@ theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow
 -/
 
 #print Sylow.card_eq_multiplicity /-
-/-- The cardinality of a Sylow group is `p ^ n`
+/-- The cardinality of a Sylow subgroup is `p ^ n`
  where `n` is the multiplicity of `p` in the group order. -/
 theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
     card P = p ^ Nat.factorization (card G) p :=
@@ -827,6 +827,27 @@ theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylo
 #align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
 -/
 
+/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
+ where `n` is the multiplicity of `p` in the group order. -/
+def ofCard [Fintype G] {p : ℕ} [hp : Fact p.Prime] (H : Subgroup G) [Fintype H]
+    (card_eq : card H = p ^ (card G).factorization p) : Sylow p G
+    where
+  toSubgroup := H
+  is_p_group' := IsPGroup.of_card card_eq
+  is_maximal' := by
+    obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
+    exact
+      SetLike.ext'
+          (Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸
+        fun _ => P.3
+#align sylow.of_card Sylow.ofCard
+
+@[simp, norm_cast]
+theorem coe_ofCard [Fintype G] {p : ℕ} [hp : Fact p.Prime] (H : Subgroup G) [Fintype H]
+    (card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
+  rfl
+#align sylow.coe_of_card Sylow.coe_ofCard
+
 #print Sylow.subsingleton_of_normal /-
 theorem subsingleton_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
     (h : (P : Subgroup G).Normal) : Subsingleton (Sylow p G) :=
@@ -904,15 +925,15 @@ theorem normal_of_normalizerCondition (hnc : NormalizerCondition G) {p : ℕ} [F
 open scoped BigOperators
 
 #print Sylow.directProductOfNormal /-
-/-- If all its sylow groups are normal, then a finite group is isomorphic to the direct product
-of these sylow groups.
+/-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
+of these Sylow subgroups.
 -/
 noncomputable def directProductOfNormal [Fintype G]
     (hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
     (∀ p : (card G).factorization.support, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G :=
   by
   set ps := (Fintype.card G).factorization.support
-  -- “The” sylow group for p
+  -- “The” Sylow subgroup for p
   let P : ∀ p, Sylow p G := default
   have hcomm : Pairwise fun p₁ p₂ : ps => ∀ x y : G, x ∈ P p₁ → y ∈ P p₂ → Commute x y :=
     by
@@ -923,7 +944,7 @@ noncomputable def directProductOfNormal [Fintype G]
     apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P p₁)) (hn (P p₂))
     apply IsPGroup.disjoint_of_ne p₁ p₂ hne' _ _ (P p₁).is_p_group' (P p₂).is_p_group'
   refine' MulEquiv.trans _ _
-  -- There is only one sylow group for each p, so the inner product is trivial
+  -- There is only one Sylow subgroup for each p, so the inner product is trivial
   show (∀ p : ps, ∀ P : Sylow p G, P) ≃* ∀ p : ps, P p
   · -- here we need to help the elaborator with an explicit instantiation
     apply @MulEquiv.piCongrRight ps (fun p => ∀ P : Sylow p G, P) (fun p => P p) _ _

bump to nightly-2023-06-13-21

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

829520ac

Diff

@@ -99,13 +99,16 @@ instance : SubgroupClass (Sylow p G) G
 
 variable (P : Sylow p G)
 
+#print Sylow.mulActionLeft /-
 /-- The action by a Sylow subgroup is the action by the underlying group. -/
 instance mulActionLeft {α : Type _} [MulAction G α] : MulAction P α :=
   Subgroup.mulAction ↑P
 #align sylow.mul_action_left Sylow.mulActionLeft
+-/
 
 variable {K : Type _} [Group K] (ϕ : K →* G) {N : Subgroup G}
 
+#print Sylow.comapOfKerIsPGroup /-
 /-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
 def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
   { P.1.comap ϕ with
@@ -115,43 +118,57 @@ def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow
       rw [← P.3 (hQ.map ϕ) (le_trans (ge_of_eq (map_comap_eq_self h)) (map_mono hle))]
       exact (comap_map_eq_self ((P.1.ker_le_comap ϕ).trans hle)).symm }
 #align sylow.comap_of_ker_is_p_group Sylow.comapOfKerIsPGroup
+-/
 
+#print Sylow.coe_comapOfKerIsPGroup /-
 @[simp]
 theorem coe_comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) :
     ↑(P.comap_of_ker_isPGroup ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
   rfl
 #align sylow.coe_comap_of_ker_is_p_group Sylow.coe_comapOfKerIsPGroup
+-/
 
+#print Sylow.comapOfInjective /-
 /-- The preimage of a Sylow subgroup under an injective homomorphism is a Sylow subgroup. -/
 def comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) : Sylow p K :=
   P.comap_of_ker_isPGroup ϕ (IsPGroup.ker_isPGroup_of_injective hϕ) h
 #align sylow.comap_of_injective Sylow.comapOfInjective
+-/
 
+#print Sylow.coe_comapOfInjective /-
 @[simp]
 theorem coe_comapOfInjective (hϕ : Function.Injective ϕ) (h : ↑P ≤ ϕ.range) :
     ↑(P.comap_of_injective ϕ hϕ h) = Subgroup.comap ϕ ↑P :=
   rfl
 #align sylow.coe_comap_of_injective Sylow.coe_comapOfInjective
+-/
 
+#print Sylow.subtype /-
 /-- A sylow subgroup of G is also a sylow subgroup of a subgroup of G. -/
 protected def subtype (h : ↑P ≤ N) : Sylow p N :=
   P.comap_of_injective N.Subtype Subtype.coe_injective (by rwa [subtype_range])
 #align sylow.subtype Sylow.subtype
+-/
 
+#print Sylow.coe_subtype /-
 @[simp]
 theorem coe_subtype (h : ↑P ≤ N) : ↑(P.Subtype h) = subgroupOf (↑P) N :=
   rfl
 #align sylow.coe_subtype Sylow.coe_subtype
+-/
 
+#print Sylow.subtype_injective /-
 theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
     (h : P.Subtype hP = Q.Subtype hQ) : P = Q :=
   by
   rw [SetLike.ext_iff] at h ⊢
   exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
 #align sylow.subtype_injective Sylow.subtype_injective
+-/
 
 end Sylow
 
+#print IsPGroup.exists_le_sylow /-
 /-- A generalization of **Sylow's first theorem**.
   Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
 theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
@@ -171,6 +188,7 @@ theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q :
       P hP)
     fun Q ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3⟩, hQ2⟩
 #align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
+-/
 
 #print Sylow.nonempty /-
 instance Sylow.nonempty : Nonempty (Sylow p G) :=
@@ -184,22 +202,29 @@ noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
 #align sylow.inhabited Sylow.inhabited
 -/
 
+#print Sylow.exists_comap_eq_of_ker_isPGroup /-
 theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type _} [Group H] (P : Sylow p H) {f : H →* G}
     (hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
   Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
     (P.2.map f).exists_le_sylow
 #align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
+-/
 
+#print Sylow.exists_comap_eq_of_injective /-
 theorem Sylow.exists_comap_eq_of_injective {H : Type _} [Group H] (P : Sylow p H) {f : H →* G}
     (hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
   P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
 #align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
+-/
 
+#print Sylow.exists_comap_subtype_eq /-
 theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
     ∃ Q : Sylow p G, (Q : Subgroup G).comap H.Subtype = P :=
   P.exists_comap_eq_of_injective Subtype.coe_injective
 #align sylow.exists_comap_subtype_eq Sylow.exists_comap_subtype_eq
+-/
 
+#print Sylow.fintypeOfKerIsPGroup /-
 /-- If the kernel of `f : H →* G` is a `p`-group,
   then `fintype (sylow p G)` implies `fintype (sylow p H)`. -/
 noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type _} [Group H] {f : H →* G}
@@ -209,12 +234,15 @@ noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type _} [Group H] {f : H →*
   let hg : ∀ P : Sylow p H, (g P).1.comap f = P := fun P => Classical.choose_spec (h_exists P)
   Fintype.ofInjective g fun P Q h => Sylow.ext (by simp only [← hg, h])
 #align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
+-/
 
+#print Sylow.fintypeOfInjective /-
 /-- If `f : H →* G` is injective, then `fintype (sylow p G)` implies `fintype (sylow p H)`. -/
 noncomputable def Sylow.fintypeOfInjective {H : Type _} [Group H] {f : H →* G}
     (hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
   Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
 #align sylow.fintype_of_injective Sylow.fintypeOfInjective
+-/
 
 /-- If `H` is a subgroup of `G`, then `fintype (sylow p G)` implies `fintype (sylow p H)`. -/
 noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
@@ -255,28 +283,39 @@ instance Sylow.mulAction : MulAction G (Sylow p G) :=
 #align sylow.mul_action Sylow.mulAction
 -/
 
+#print Sylow.smul_def /-
 theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
   rfl
 #align sylow.smul_def Sylow.smul_def
+-/
 
+#print Sylow.coe_subgroup_smul /-
 theorem Sylow.coe_subgroup_smul {g : G} {P : Sylow p G} :
     ↑(g • P) = MulAut.conj g • (P : Subgroup G) :=
   rfl
 #align sylow.coe_subgroup_smul Sylow.coe_subgroup_smul
+-/
 
+#print Sylow.coe_smul /-
 theorem Sylow.coe_smul {g : G} {P : Sylow p G} : ↑(g • P) = MulAut.conj g • (P : Set G) :=
   rfl
 #align sylow.coe_smul Sylow.coe_smul
+-/
 
+#print Sylow.smul_le /-
 theorem Sylow.smul_le {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) : ↑(h • P) ≤ H :=
   Subgroup.conj_smul_le_of_le hP h
 #align sylow.smul_le Sylow.smul_le
+-/
 
+#print Sylow.smul_subtype /-
 theorem Sylow.smul_subtype {P : Sylow p G} {H : Subgroup G} (hP : ↑P ≤ H) (h : H) :
     h • P.Subtype hP = (h • P).Subtype (Sylow.smul_le hP h) :=
   Sylow.ext (Subgroup.conj_smul_subgroupOf hP h)
 #align sylow.smul_subtype Sylow.smul_subtype
+-/
 
+#print Sylow.smul_eq_iff_mem_normalizer /-
 theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
     g • P = P ↔ g ∈ (P : Subgroup G).normalizer :=
   by
@@ -289,6 +328,7 @@ theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
             ((congr_arg (· ∈ P.1) (MulAut.inv_apply_self G (MulAut.conj g) a)).mpr b),
           fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
 #align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
+-/
 
 #print Sylow.smul_eq_of_normal /-
 theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
@@ -296,11 +336,14 @@ theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).No
 #align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
 -/
 
+#print Subgroup.sylow_mem_fixedPoints_iff /-
 theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
     P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
   simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer] <;> exact Subtype.forall
 #align subgroup.sylow_mem_fixed_points_iff Subgroup.sylow_mem_fixedPoints_iff
+-/
 
+#print IsPGroup.inf_normalizer_sylow /-
 theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q : Sylow p G) :
     P ⊓ (Q : Subgroup G).normalizer = P ⊓ Q :=
   le_antisymm
@@ -309,11 +352,14 @@ theorem IsPGroup.inf_normalizer_sylow {P : Subgroup G} (hP : IsPGroup p P) (Q :
         (Q.3 (hP.to_inf_left.to_sup_of_normal_right' Q.2 inf_le_right) le_sup_right)))
     (inf_le_inf_left P le_normalizer)
 #align is_p_group.inf_normalizer_sylow IsPGroup.inf_normalizer_sylow
+-/
 
+#print IsPGroup.sylow_mem_fixedPoints_iff /-
 theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P) {Q : Sylow p G} :
     Q ∈ fixedPoints P (Sylow p G) ↔ P ≤ Q := by
   rw [P.sylow_mem_fixed_points_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
 #align is_p_group.sylow_mem_fixed_points_iff IsPGroup.sylow_mem_fixedPoints_iff
+-/
 
 /-- A generalization of **Sylow's second theorem**.
   If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate. -/
@@ -372,22 +418,28 @@ theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ c
 
 variable {p} {G}
 
+#print Sylow.equivSMul /-
 /-- Sylow subgroups are isomorphic -/
 def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
   equivSMul (MulAut.conj g) ↑P
 #align sylow.equiv_smul Sylow.equivSMul
+-/
 
+#print Sylow.equiv /-
 /-- Sylow subgroups are isomorphic -/
 noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q :=
   by
   rw [← Classical.choose_spec (exists_smul_eq G P Q)]
   exact P.equiv_smul (Classical.choose (exists_smul_eq G P Q))
 #align sylow.equiv Sylow.equiv
+-/
 
+#print Sylow.orbit_eq_top /-
 @[simp]
 theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) : orbit G P = ⊤ :=
   top_le_iff.mp fun Q hQ => exists_smul_eq G P Q
 #align sylow.orbit_eq_top Sylow.orbit_eq_top
+-/
 
 #print Sylow.stabilizer_eq_normalizer /-
 theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
@@ -396,6 +448,7 @@ theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
 #align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
 -/
 
+#print Sylow.conj_eq_normalizer_conj_of_mem_centralizer /-
 theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
     (P : Sylow p G) (x g : G) (hx : x ∈ (P : Subgroup G).centralizer)
     (hy : g⁻¹ * x * g ∈ (P : Subgroup G).centralizer) :
@@ -414,12 +467,15 @@ theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite
   refine' ⟨h * g, sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
   rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
 #align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
+-/
 
+#print Sylow.conj_eq_normalizer_conj_of_mem /-
 theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
     [hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
     ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
   P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
 #align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
+-/
 
 #print Sylow.equivQuotientNormalizer /-
 /-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
@@ -495,6 +551,7 @@ theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow
 #align not_dvd_index_sylow not_dvd_index_sylow
 -/
 
+#print Sylow.normalizer_sup_eq_top /-
 /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
   of `N`, then `N_G(P) ⊔ N = G`. -/
 theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
@@ -514,7 +571,9 @@ theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N
   rw [map_map, ← congr_arg (map N.subtype) hn, map_map]
   rfl
 #align sylow.normalizer_sup_eq_top Sylow.normalizer_sup_eq_top
+-/
 
+#print Sylow.normalizer_sup_eq_top' /-
 /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
   of `N`, then `N_G(P) ⊔ N = G`. -/
 theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
@@ -522,6 +581,7 @@ theorem Sylow.normalizer_sup_eq_top' {p : ℕ} [Fact p.Prime] {N : Subgroup G} [
   rw [← Sylow.normalizer_sup_eq_top (P.subtype hP), P.coe_subtype, subgroup_of_map_subtype,
     inf_of_le_left hP]
 #align sylow.normalizer_sup_eq_top' Sylow.normalizer_sup_eq_top'
+-/
 
 end InfiniteSylow
 
@@ -546,6 +606,7 @@ namespace Sylow
 
 open Subgroup Submonoid MulAction
 
+#print Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer /-
 theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite ↥(H : Set G)]
     {x : G} : (x : G ⧸ H) ∈ fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
   ⟨fun hx =>
@@ -565,7 +626,9 @@ theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Fin
               (mul_mem_cancel_left (inv_mem hb₁)).1 <| by
                 rw [hx] at hb₂  <;> simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
 #align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
+-/
 
+#print Sylow.fixedPointsMulLeftCosetsEquivQuotient /-
 /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
 def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)] :
     MulAction.fixedPoints H (G ⧸ H) ≃
@@ -574,7 +637,9 @@ def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)]
     (fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
     (by intros; rw [setoidHasEquiv]; simp only [left_rel_apply]; rfl)
 #align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
+-/
 
+#print Sylow.card_quotient_normalizer_modEq_card_quotient /-
 /-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
   mod `p` to the index of `H`.  -/
 theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
@@ -585,7 +650,9 @@ theorem card_quotient_normalizer_modEq_card_quotient [Fintype G] {p : ℕ} {n :
   rw [← Fintype.card_congr (fixed_points_mul_left_cosets_equiv_quotient H)]
   exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
 #align sylow.card_quotient_normalizer_modeq_card_quotient Sylow.card_quotient_normalizer_modEq_card_quotient
+-/
 
+#print Sylow.card_normalizer_modEq_card /-
 /-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
   normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`.  -/
 theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
@@ -597,7 +664,9 @@ theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.
     hH, pow_succ]
   exact (card_quotient_normalizer_modeq_card_quotient hH).mul_right' _
 #align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
+-/
 
+#print Sylow.prime_dvd_card_quotient_normalizer /-
 /-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup, then `p` divides the
   index of `H` inside its normalizer. -/
 theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
@@ -614,7 +683,9 @@ theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp :
     hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
   Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm )
 #align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
+-/
 
+#print Sylow.prime_pow_dvd_card_normalizer /-
 /-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
   then `p ^ (n + 1)` divides the cardinality of the normalizer of `H`. -/
 theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
@@ -622,7 +693,9 @@ theorem prime_pow_dvd_card_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact
     p ^ (n + 1) ∣ card (normalizer H) :=
   Nat.modEq_zero_iff_dvd.1 ((card_normalizer_modEq_card hH).trans hdvd.modEq_zero_nat)
 #align sylow.prime_pow_dvd_card_normalizer Sylow.prime_pow_dvd_card_normalizer
+-/
 
+#print Sylow.exists_subgroup_card_pow_succ /-
 /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
   then `H` is contained in a subgroup of cardinality `p ^ (n + 1)`
   if `p ^ (n + 1)` divides the cardinality of `G` -/
@@ -668,7 +741,9 @@ theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact
     rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
     simpa using hy⟩
 #align sylow.exists_subgroup_card_pow_succ Sylow.exists_subgroup_card_pow_succ
+-/
 
+#print Sylow.exists_subgroup_card_pow_prime_le /-
 /-- If `H` is a subgroup of `G` of cardinality `p ^ n`,
   then `H` is contained in a subgroup of cardinality `p ^ m`
   if `n ≤ m` and `p ^ m` divides the cardinality of `G` -/
@@ -689,7 +764,9 @@ theorem exists_subgroup_card_pow_prime_le [Fintype G] (p : ℕ) :
         ⟨K', by rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le], le_trans hK.2 hK'.2⟩)
       fun hnm : n = m => ⟨H, by simp [hH, hnm]⟩
 #align sylow.exists_subgroup_card_pow_prime_le Sylow.exists_subgroup_card_pow_prime_le
+-/
 
+#print Sylow.exists_subgroup_card_pow_prime /-
 /-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
   the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
 theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.Prime]
@@ -697,14 +774,18 @@ theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.P
   let ⟨K, hK⟩ := exists_subgroup_card_pow_prime_le p hdvd ⊥ (by simp) n.zero_le
   ⟨K, hK.1⟩
 #align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
+-/
 
+#print Sylow.pow_dvd_card_of_pow_dvd_card /-
 theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
     (hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
   (hp.1.coprime_pow_of_not_dvd
           (not_dvd_index_sylow P index_ne_zero_of_finite)).symm.dvd_of_dvd_mul_left
     ((index_mul_card P.1).symm ▸ hdvd)
 #align sylow.pow_dvd_card_of_pow_dvd_card Sylow.pow_dvd_card_of_pow_dvd_card
+-/
 
+#print Sylow.dvd_card_of_dvd_card /-
 theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
     (hdvd : p ∣ card G) : p ∣ card P :=
   by
@@ -712,14 +793,18 @@ theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G
   have key := P.pow_dvd_card_of_pow_dvd_card hdvd
   rwa [pow_one] at key 
 #align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
+-/
 
+#print Sylow.card_coprime_index /-
 /-- Sylow subgroups are Hall subgroups. -/
 theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
     (card P).coprime (index (P : Subgroup G)) :=
   let ⟨n, hn⟩ := IsPGroup.iff_card.mp P.2
   hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
 #align sylow.card_coprime_index Sylow.card_coprime_index
+-/
 
+#print Sylow.ne_bot_of_dvd_card /-
 theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
     (hdvd : p ∣ card G) : (P : Subgroup G) ≠ ⊥ :=
   by
@@ -727,7 +812,9 @@ theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow
   have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
   rwa [h, card_bot] at key 
 #align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
+-/
 
+#print Sylow.card_eq_multiplicity /-
 /-- The cardinality of a Sylow group is `p ^ n`
  where `n` is the multiplicity of `p` in the group order. -/
 theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
@@ -738,6 +825,7 @@ theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylo
   rw [HEq, ← hp.out.pow_dvd_iff_dvd_ord_proj (show card G ≠ 0 from card_ne_zero), ← HEq]
   exact P.1.card_subgroup_dvd_card
 #align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
+-/
 
 #print Sylow.subsingleton_of_normal /-
 theorem subsingleton_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
@@ -790,6 +878,7 @@ theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P :
 #align sylow.normalizer_normalizer Sylow.normalizer_normalizer
 -/
 
+#print Sylow.normal_of_all_max_subgroups_normal /-
 theorem normal_of_all_max_subgroups_normal [Finite G]
     (hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
     (P : Sylow p G) : (↑P : Subgroup G).Normal :=
@@ -802,6 +891,7 @@ theorem normal_of_all_max_subgroups_normal [Finite G]
         refine' (hK.1 _).elim
         rw [← sup_of_le_right hNK, P.normalizer_sup_eq_top' hPK])
 #align sylow.normal_of_all_max_subgroups_normal Sylow.normal_of_all_max_subgroups_normal
+-/
 
 #print Sylow.normal_of_normalizerCondition /-
 theorem normal_of_normalizerCondition (hnc : NormalizerCondition G) {p : ℕ} [Fact p.Prime]
@@ -813,6 +903,7 @@ theorem normal_of_normalizerCondition (hnc : NormalizerCondition G) {p : ℕ} [F
 
 open scoped BigOperators
 
+#print Sylow.directProductOfNormal /-
 /-- If all its sylow groups are normal, then a finite group is isomorphic to the direct product
 of these sylow groups.
 -/
@@ -866,6 +957,7 @@ noncomputable def directProductOfNormal [Fintype G]
       _ = (card G).factorization.Prod pow := rfl
       _ = card G := Nat.factorization_prod_pow_eq_self Fintype.card_ne_zero
 #align sylow.direct_product_of_normal Sylow.directProductOfNormal
+-/
 
 end Sylow

bump to nightly-2023-06-09-07

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

16afdc39

Diff

@@ -327,7 +327,6 @@ instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G
           forall_congr' fun a => Subtype.ext_iff
         _ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
         _ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
-        
     suffices Set.Nonempty (fixed_points Q (orbit G P)) by
       exact Exists.elim this fun R hR => (congr_arg _ (Sylow.ext (H.mp hR))).mp R.2
     apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
@@ -336,7 +335,6 @@ instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G
       1 = card (fixed_points P (orbit G P)) := _
       _ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
       _ ≡ 0 [MOD p] := nat.modeq_zero_iff_dvd.mpr h
-      
     rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
     refine' card_congr' (congr_arg _ (Eq.symm _))
     rw [Set.eq_singleton_iff_unique_mem]
@@ -356,7 +354,6 @@ theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] : card (Sylow
         Q ∈ fixed_points P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff
         _ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
         _ ↔ Q ∈ {P} := sylow.ext_iff.symm.trans set.mem_singleton_iff.symm
-        
   have : Fintype (fixed_points P.1 (Sylow p G)) := by rw [this]; infer_instance
   have : card (fixed_points P.1 (Sylow p G)) = 1 := by simp [this]
   exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
@@ -433,7 +430,6 @@ noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p
     _ ≃ orbit G P := by rw [P.orbit_eq_top]
     _ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
     _ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
-    
 #align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
 -/
 
@@ -869,7 +865,6 @@ noncomputable def directProductOfNormal [Fintype G]
         (Finset.prod_finset_coe (fun p => p ^ (card G).factorization p) _)
       _ = (card G).factorization.Prod pow := rfl
       _ = card G := Nat.factorization_prod_pow_eq_self Fintype.card_ne_zero
-      
 #align sylow.direct_product_of_normal Sylow.directProductOfNormal
 
 end Sylow

bump to nightly-2023-06-04-18

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

3fb4268b

Diff

@@ -156,7 +156,7 @@ end Sylow
   Every `p`-subgroup is contained in a Sylow `p`-subgroup. -/
 theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q : Sylow p G, P ≤ Q :=
   Exists.elim
-    (zorn_nonempty_partialOrder₀ { Q : Subgroup G | IsPGroup p Q }
+    (zorn_nonempty_partialOrder₀ {Q : Subgroup G | IsPGroup p Q}
       (fun c hc1 hc2 Q hQ =>
         ⟨{  carrier := ⋃ R : c, R
             one_mem' := ⟨Q, ⟨⟨Q, hQ⟩, rfl⟩, Q.one_mem⟩
@@ -320,27 +320,27 @@ theorem IsPGroup.sylow_mem_fixedPoints_iff {P : Subgroup G} (hP : IsPGroup p P)
 instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G) :=
   ⟨fun P Q => by
     classical
-      cases nonempty_fintype (Sylow p G)
-      have H := fun {R : Sylow p G} {S : orbit G P} =>
-        calc
-          S ∈ fixed_points R (orbit G P) ↔ S.1 ∈ fixed_points R (Sylow p G) :=
-            forall_congr' fun a => Subtype.ext_iff
-          _ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
-          _ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
-          
-      suffices Set.Nonempty (fixed_points Q (orbit G P)) by
-        exact Exists.elim this fun R hR => (congr_arg _ (Sylow.ext (H.mp hR))).mp R.2
-      apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
-      refine' fun h => hp.out.not_dvd_one (nat.modeq_zero_iff_dvd.mp _)
+    cases nonempty_fintype (Sylow p G)
+    have H := fun {R : Sylow p G} {S : orbit G P} =>
       calc
-        1 = card (fixed_points P (orbit G P)) := _
-        _ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
-        _ ≡ 0 [MOD p] := nat.modeq_zero_iff_dvd.mpr h
+        S ∈ fixed_points R (orbit G P) ↔ S.1 ∈ fixed_points R (Sylow p G) :=
+          forall_congr' fun a => Subtype.ext_iff
+        _ ↔ R.1 ≤ S := R.2.sylow_mem_fixedPoints_iff
+        _ ↔ S.1.1 = R := ⟨fun h => R.3 S.1.2 h, ge_of_eq⟩
         
-      rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
-      refine' card_congr' (congr_arg _ (Eq.symm _))
-      rw [Set.eq_singleton_iff_unique_mem]
-      exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
+    suffices Set.Nonempty (fixed_points Q (orbit G P)) by
+      exact Exists.elim this fun R hR => (congr_arg _ (Sylow.ext (H.mp hR))).mp R.2
+    apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
+    refine' fun h => hp.out.not_dvd_one (nat.modeq_zero_iff_dvd.mp _)
+    calc
+      1 = card (fixed_points P (orbit G P)) := _
+      _ ≡ card (orbit G P) [MOD p] := (P.2.card_modEq_card_fixedPoints (orbit G P)).symm
+      _ ≡ 0 [MOD p] := nat.modeq_zero_iff_dvd.mpr h
+      
+    rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G P)]
+    refine' card_congr' (congr_arg _ (Eq.symm _))
+    rw [Set.eq_singleton_iff_unique_mem]
+    exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩⟩
 
 variable (p) (G)

bump to nightly-2023-06-01-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

b9c87c70

Diff

@@ -146,7 +146,7 @@ theorem coe_subtype (h : ↑P ≤ N) : ↑(P.Subtype h) = subgroupOf (↑P) N :=
 theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
     (h : P.Subtype hP = Q.Subtype hQ) : P = Q :=
   by
-  rw [SetLike.ext_iff] at h⊢
+  rw [SetLike.ext_iff] at h ⊢
   exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
 #align sylow.subtype_injective Sylow.subtype_injective
 
@@ -167,7 +167,7 @@ theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q :
           fun ⟨g, _, ⟨S, rfl⟩, hg⟩ =>
           by
           refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
-          rwa [Subtype.ext_iff, coe_pow] at hk⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
+          rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
       P hP)
     fun Q ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3⟩, hQ2⟩
 #align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
@@ -411,9 +411,9 @@ theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite
     rintro - ⟨z, hz, rfl⟩
     specialize hy z hz
     rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
-      eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
+      eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy 
   obtain ⟨h, hh⟩ := exists_smul_eq (zpowers x).centralizer ((g • P).Subtype h2) (P.subtype h1)
-  simp_rw [Sylow.smul_subtype, smul_def, smul_smul] at hh
+  simp_rw [Sylow.smul_subtype, smul_def, smul_smul] at hh 
   refine' ⟨h * g, sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
   rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]
 #align sylow.conj_eq_normalizer_conj_of_mem_centralizer Sylow.conj_eq_normalizer_conj_of_mem_centralizer
@@ -469,7 +469,7 @@ theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup
   by
   intro h
   haveI := (P : Subgroup G).fintypeQuotientOfFiniteIndex
-  rw [index_eq_card] at h
+  rw [index_eq_card] at h 
   obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card p h
   have h := IsPGroup.of_card ((order_eq_card_zpowers.symm.trans hx).trans (pow_one p).symm)
   let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
@@ -480,7 +480,7 @@ theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup
   replace hp := mt order_of_eq_one_iff.mpr (ne_of_eq_of_ne hx hp.1.ne_one)
   rw [← zpowers_eq_bot, ← Ne, ← bot_lt_iff_ne_bot, ←
     comap_lt_comap_of_surjective (QuotientGroup.mk'_surjective _), MonoidHom.comap_bot,
-    QuotientGroup.ker_mk'] at hp
+    QuotientGroup.ker_mk'] at hp 
   exact hp.ne' (P.3 hQ hp.le)
 #align not_dvd_index_sylow' not_dvd_index_sylow'
 -/
@@ -509,7 +509,7 @@ theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N
   rw [← inv_mul_cancel_left (↑n) g, sup_comm]
   apply mul_mem_sup (N.inv_mem n.2)
   rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← mul_aut.conj_normal.map_mul,
-    Sylow.ext_iff, Sylow.pointwise_smul_def, pointwise_smul_def] at hn
+    Sylow.ext_iff, Sylow.pointwise_smul_def, pointwise_smul_def] at hn 
   refine' fun x =>
     (mem_map_iff_mem
             (show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from
@@ -557,7 +557,7 @@ theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Fin
     inv_mem_iff.1
       (mem_normalizer_fintype fun n (hn : n ∈ H) =>
         have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha (mem_orbit _ ⟨n⁻¹, H.inv_mem hn⟩))
-        show _ ∈ H by rw [mul_inv_rev, inv_inv] at this; convert this; rw [inv_inv]),
+        show _ ∈ H by rw [mul_inv_rev, inv_inv] at this ; convert this; rw [inv_inv]),
     fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
     (mem_fixedPoints' _).2 fun y =>
       Quotient.inductionOn' y fun y hy =>
@@ -567,7 +567,7 @@ theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Fin
           inv_mem_iff.1 <|
             (hx _).2 <|
               (mul_mem_cancel_left (inv_mem hb₁)).1 <| by
-                rw [hx] at hb₂ <;> simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
+                rw [hx] at hb₂  <;> simpa [mul_inv_rev, mul_assoc] using hb₂)⟩
 #align sylow.mem_fixed_points_mul_left_cosets_iff_mem_normalizer Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer
 
 /-- The fixed points of the action of `H` on its cosets correspond to `normalizer H / H`. -/
@@ -576,7 +576,7 @@ def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)]
       normalizer H ⧸ Subgroup.comap ((normalizer H).Subtype : normalizer H →* G) H :=
   @subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (id _) (id _) (fixedPoints _ _)
     (fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
-    (by intros ; rw [setoidHasEquiv]; simp only [left_rel_apply]; rfl)
+    (by intros; rw [setoidHasEquiv]; simp only [left_rel_apply]; rfl)
 #align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
 
 /-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
@@ -616,7 +616,7 @@ theorem prime_dvd_card_quotient_normalizer [Fintype G] {p : ℕ} {n : ℕ} [hp :
     s * p % p =
       card (normalizer H ⧸ Subgroup.comap ((normalizer H).Subtype : normalizer H →* G) H) % p :=
     hcard ▸ (card_quotient_normalizer_modEq_card_quotient hH).symm
-  Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
+  Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm )
 #align sylow.prime_dvd_card_quotient_normalizer Sylow.prime_dvd_card_quotient_normalizer
 
 /-- If `H` is a `p`-subgroup but not a Sylow `p`-subgroup of cardinality `p ^ n`,
@@ -642,7 +642,7 @@ theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact
     card_congr (fixedPointsMulLeftCosetsEquivQuotient H) ▸
       hcard ▸ (IsPGroup.of_card hH).card_modEq_card_fixedPoints _
   have hm' : p ∣ card (normalizer H ⧸ H.subgroupOf H.normalizer) :=
-    Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm)
+    Nat.dvd_of_mod_eq_zero (by rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm )
   let ⟨x, hx⟩ := @exists_prime_orderOf_dvd_card _ (QuotientGroup.Quotient.group _) _ _ hp hm'
   have hequiv : H ≃ H.subgroupOf H.normalizer := (subgroupOfEquivOfLe le_normalizer).symm.toEquiv
   ⟨Subgroup.map (normalizer H).Subtype
@@ -712,9 +712,9 @@ theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime]
 theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G)
     (hdvd : p ∣ card G) : p ∣ card P :=
   by
-  rw [← pow_one p] at hdvd
+  rw [← pow_one p] at hdvd 
   have key := P.pow_dvd_card_of_pow_dvd_card hdvd
-  rwa [pow_one] at key
+  rwa [pow_one] at key 
 #align sylow.dvd_card_of_dvd_card Sylow.dvd_card_of_dvd_card
 
 /-- Sylow subgroups are Hall subgroups. -/
@@ -729,7 +729,7 @@ theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow
   by
   refine' fun h => hp.out.not_dvd_one _
   have key : p ∣ card (P : Subgroup G) := P.dvd_card_of_dvd_card hdvd
-  rwa [h, card_bot] at key
+  rwa [h, card_bot] at key 
 #align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
 
 /-- The cardinality of a Sylow group is `p ^ n`
@@ -751,7 +751,7 @@ theorem subsingleton_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P
   intro Q R
   obtain ⟨x, h1⟩ := exists_smul_eq G P Q
   obtain ⟨x, h2⟩ := exists_smul_eq G P R
-  rw [Sylow.smul_eq_of_normal] at h1 h2
+  rw [Sylow.smul_eq_of_normal] at h1 h2 
   rw [← h1, ← h2]
 #align sylow.subsingleton_of_normal Sylow.subsingleton_of_normal
 -/
@@ -788,7 +788,7 @@ theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P :
   by
   have := normal_of_normalizer_normal (P.subtype (le_normalizer.trans le_normalizer))
   simp_rw [← normalizer_eq_top, coeSubtype, ← subgroup_of_normalizer_eq le_normalizer, ←
-    subgroup_of_normalizer_eq le_rfl, subgroup_of_self] at this
+    subgroup_of_normalizer_eq le_rfl, subgroup_of_self] at this 
   rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map, ← this rfl]
   exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))
 #align sylow.normalizer_normalizer Sylow.normalizer_normalizer

bump to nightly-2023-06-01-04

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

4d9eaa85

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Thomas Browning
 
 ! This file was ported from Lean 3 source module group_theory.sylow
-! leanprover-community/mathlib commit c55911f6166b348e1c72b08c8664e3af5f1ce334
+! leanprover-community/mathlib commit f60c6087a7275b72d5db3c5a1d0e19e35a429c0a
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -18,6 +18,9 @@ import Mathbin.Order.Atoms.Finite
 /-!
 # Sylow theorems
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 The Sylow theorems are the following results for every finite group `G` and every prime number `p`.
 
 * There exists a Sylow `p`-subgroup of `G`.
@@ -52,11 +55,13 @@ section InfiniteSylow
 
 variable (p : ℕ) (G : Type _) [Group G]
 
+#print Sylow /-
 /-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
 structure Sylow extends Subgroup G where
   is_p_group' : IsPGroup p to_subgroup
   is_maximal' : ∀ {Q : Subgroup G}, IsPGroup p Q → to_subgroup ≤ Q → Q = to_subgroup
 #align sylow Sylow
+-/
 
 variable {p} {G}
 
@@ -70,13 +75,17 @@ theorem toSubgroup_eq_coe {P : Sylow p G} : P.toSubgroup = ↑P :=
   rfl
 #align sylow.to_subgroup_eq_coe Sylow.toSubgroup_eq_coe
 
+#print Sylow.ext /-
 @[ext]
 theorem ext {P Q : Sylow p G} (h : (P : Subgroup G) = Q) : P = Q := by cases P <;> cases Q <;> congr
 #align sylow.ext Sylow.ext
+-/
 
+#print Sylow.ext_iff /-
 theorem ext_iff {P Q : Sylow p G} : P = Q ↔ (P : Subgroup G) = Q :=
   ⟨congr_arg coe, ext⟩
 #align sylow.ext_iff Sylow.ext_iff
+-/
 
 instance : SetLike (Sylow p G) G where
   coe := coe
@@ -163,13 +172,17 @@ theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q :
     fun Q ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3⟩, hQ2⟩
 #align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow
 
+#print Sylow.nonempty /-
 instance Sylow.nonempty : Nonempty (Sylow p G) :=
   nonempty_of_exists IsPGroup.of_bot.exists_le_sylow
 #align sylow.nonempty Sylow.nonempty
+-/
 
+#print Sylow.inhabited /-
 noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
   Classical.inhabited_of_nonempty Sylow.nonempty
 #align sylow.inhabited Sylow.inhabited
+-/
 
 theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type _} [Group H] (P : Sylow p H) {f : H →* G}
     (hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
@@ -213,6 +226,7 @@ instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
 
 open scoped Pointwise
 
+#print Sylow.pointwiseMulAction /-
 /-- `subgroup.pointwise_mul_action` preserves Sylow subgroups. -/
 instance Sylow.pointwiseMulAction {α : Type _} [Group α] [MulDistribMulAction α G] :
     MulAction α (Sylow p G)
@@ -226,15 +240,20 @@ instance Sylow.pointwiseMulAction {α : Type _} [Group α] [MulDistribMulAction
   one_smul P := Sylow.ext (one_smul α P)
   mul_smul g h P := Sylow.ext (mul_smul g h P)
 #align sylow.pointwise_mul_action Sylow.pointwiseMulAction
+-/
 
+#print Sylow.pointwise_smul_def /-
 theorem Sylow.pointwise_smul_def {α : Type _} [Group α] [MulDistribMulAction α G] {g : α}
     {P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
   rfl
 #align sylow.pointwise_smul_def Sylow.pointwise_smul_def
+-/
 
+#print Sylow.mulAction /-
 instance Sylow.mulAction : MulAction G (Sylow p G) :=
   compHom _ MulAut.conj
 #align sylow.mul_action Sylow.mulAction
+-/
 
 theorem Sylow.smul_def {g : G} {P : Sylow p G} : g • P = MulAut.conj g • P :=
   rfl
@@ -271,9 +290,11 @@ theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
           fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
 #align sylow.smul_eq_iff_mem_normalizer Sylow.smul_eq_iff_mem_normalizer
 
+#print Sylow.smul_eq_of_normal /-
 theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
   by simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
 #align sylow.smul_eq_of_normal Sylow.smul_eq_of_normal
+-/
 
 theorem Subgroup.sylow_mem_fixedPoints_iff (H : Subgroup G) {P : Sylow p G} :
     P ∈ fixedPoints H (Sylow p G) ↔ H ≤ (P : Subgroup G).normalizer := by
@@ -323,6 +344,7 @@ instance [hp : Fact p.Prime] [Finite (Sylow p G)] : IsPretransitive G (Sylow p G
 
 variable (p) (G)
 
+#print card_sylow_modEq_one /-
 /-- A generalization of **Sylow's third theorem**.
   If the number of Sylow `p`-subgroups is finite, then it is congruent to `1` modulo `p`. -/
 theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] : card (Sylow p G) ≡ 1 [MOD p] :=
@@ -339,7 +361,9 @@ theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] : card (Sylow
   have : card (fixed_points P.1 (Sylow p G)) = 1 := by simp [this]
   exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
 #align card_sylow_modeq_one card_sylow_modEq_one
+-/
 
+#print not_dvd_card_sylow /-
 theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ card (Sylow p G) :=
   fun h =>
   hp.1.ne_one
@@ -347,13 +371,14 @@ theorem not_dvd_card_sylow [hp : Fact p.Prime] [Fintype (Sylow p G)] : ¬p ∣ c
       ((Nat.modEq_iff_dvd' zero_le_one).mp
         ((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
 #align not_dvd_card_sylow not_dvd_card_sylow
+-/
 
 variable {p} {G}
 
 /-- Sylow subgroups are isomorphic -/
-def Sylow.equivSmul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
+def Sylow.equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
   equivSMul (MulAut.conj g) ↑P
-#align sylow.equiv_smul Sylow.equivSmul
+#align sylow.equiv_smul Sylow.equivSMul
 
 /-- Sylow subgroups are isomorphic -/
 noncomputable def Sylow.equiv [Fact p.Prime] [Finite (Sylow p G)] (P Q : Sylow p G) : P ≃* Q :=
@@ -367,10 +392,12 @@ theorem Sylow.orbit_eq_top [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
   top_le_iff.mp fun Q hQ => exists_smul_eq G P Q
 #align sylow.orbit_eq_top Sylow.orbit_eq_top
 
+#print Sylow.stabilizer_eq_normalizer /-
 theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
     stabilizer G P = (P : Subgroup G).normalizer :=
   ext fun g => Sylow.smul_eq_iff_mem_normalizer
 #align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
+-/
 
 theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
     (P : Sylow p G) (x g : G) (hx : x ∈ (P : Subgroup G).centralizer)
@@ -397,6 +424,7 @@ theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)]
   P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
 #align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
 
+#print Sylow.equivQuotientNormalizer /-
 /-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
 noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
     (P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
@@ -407,27 +435,35 @@ noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p
     _ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
     
 #align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
+-/
 
 noncomputable instance [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
     Fintype (G ⧸ (P : Subgroup G).normalizer) :=
   ofEquiv (Sylow p G) P.equivQuotientNormalizer
 
+#print card_sylow_eq_card_quotient_normalizer /-
 theorem card_sylow_eq_card_quotient_normalizer [Fact p.Prime] [Fintype (Sylow p G)]
     (P : Sylow p G) : card (Sylow p G) = card (G ⧸ (P : Subgroup G).normalizer) :=
   card_congr P.equivQuotientNormalizer
 #align card_sylow_eq_card_quotient_normalizer card_sylow_eq_card_quotient_normalizer
+-/
 
+#print card_sylow_eq_index_normalizer /-
 theorem card_sylow_eq_index_normalizer [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
     card (Sylow p G) = (P : Subgroup G).normalizer.index :=
   (card_sylow_eq_card_quotient_normalizer P).trans (P : Subgroup G).normalizer.index_eq_card.symm
 #align card_sylow_eq_index_normalizer card_sylow_eq_index_normalizer
+-/
 
+#print card_sylow_dvd_index /-
 theorem card_sylow_dvd_index [Fact p.Prime] [Fintype (Sylow p G)] (P : Sylow p G) :
     card (Sylow p G) ∣ (P : Subgroup G).index :=
   ((congr_arg _ (card_sylow_eq_index_normalizer P)).mp dvd_rfl).trans
     (index_dvd_of_le le_normalizer)
 #align card_sylow_dvd_index card_sylow_dvd_index
+-/
 
+#print not_dvd_index_sylow' /-
 theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup G).Normal]
     [FiniteIndex (P : Subgroup G)] : ¬p ∣ (P : Subgroup G).index :=
   by
@@ -447,7 +483,9 @@ theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup
     QuotientGroup.ker_mk'] at hp
   exact hp.ne' (P.3 hQ hp.le)
 #align not_dvd_index_sylow' not_dvd_index_sylow'
+-/
 
+#print not_dvd_index_sylow /-
 theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
     (hP : relindex ↑P (P : Subgroup G).normalizer ≠ 0) : ¬p ∣ (P : Subgroup G).index :=
   by
@@ -459,6 +497,7 @@ theorem not_dvd_index_sylow [hp : Fact p.Prime] [Finite (Sylow p G)] (P : Sylow
   replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)
   exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)
 #align not_dvd_index_sylow not_dvd_index_sylow
+-/
 
 /-- **Frattini's Argument**: If `N` is a normal subgroup of `G`, and if `P` is a Sylow `p`-subgroup
   of `N`, then `N_G(P) ⊔ N = G`. -/
@@ -500,10 +539,12 @@ variable {G : Type u} {α : Type v} {β : Type w} [Group G]
 
 attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
 
+#print QuotientGroup.card_preimage_mk /-
 theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
     Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by
   rw [← Fintype.card_prod, Fintype.card_congr (preimage_mk_equiv_subgroup_times_set _ _)]
 #align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
+-/
 
 namespace Sylow
 
@@ -702,6 +743,7 @@ theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylo
   exact P.1.card_subgroup_dvd_card
 #align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
 
+#print Sylow.subsingleton_of_normal /-
 theorem subsingleton_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
     (h : (P : Subgroup G).Normal) : Subsingleton (Sylow p G) :=
   by
@@ -712,11 +754,13 @@ theorem subsingleton_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P
   rw [Sylow.smul_eq_of_normal] at h1 h2
   rw [← h1, ← h2]
 #align sylow.subsingleton_of_normal Sylow.subsingleton_of_normal
+-/
 
 section Pointwise
 
 open scoped Pointwise
 
+#print Sylow.characteristic_of_normal /-
 theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
     (h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic :=
   by
@@ -726,14 +770,18 @@ theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (
   show (Φ • P).toSubgroup = P.to_subgroup
   congr
 #align sylow.characteristic_of_normal Sylow.characteristic_of_normal
+-/
 
 end Pointwise
 
+#print Sylow.normal_of_normalizer_normal /-
 theorem normal_of_normalizer_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
     (hn : (↑P : Subgroup G).normalizer.Normal) : (↑P : Subgroup G).Normal := by
   rw [← normalizer_eq_top, ← normalizer_sup_eq_top' P le_normalizer, sup_idem]
 #align sylow.normal_of_normalizer_normal Sylow.normal_of_normalizer_normal
+-/
 
+#print Sylow.normalizer_normalizer /-
 @[simp]
 theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G) :
     (↑P : Subgroup G).normalizer.normalizer = (↑P : Subgroup G).normalizer :=
@@ -744,6 +792,7 @@ theorem normalizer_normalizer {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P :
   rw [← subtype_range (P : Subgroup G).normalizer.normalizer, MonoidHom.range_eq_map, ← this rfl]
   exact map_comap_eq_self (le_normalizer.trans (ge_of_eq (subtype_range _)))
 #align sylow.normalizer_normalizer Sylow.normalizer_normalizer
+-/
 
 theorem normal_of_all_max_subgroups_normal [Finite G]
     (hnc : ∀ H : Subgroup G, IsCoatom H → H.Normal) {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)]
@@ -758,11 +807,13 @@ theorem normal_of_all_max_subgroups_normal [Finite G]
         rw [← sup_of_le_right hNK, P.normalizer_sup_eq_top' hPK])
 #align sylow.normal_of_all_max_subgroups_normal Sylow.normal_of_all_max_subgroups_normal
 
+#print Sylow.normal_of_normalizerCondition /-
 theorem normal_of_normalizerCondition (hnc : NormalizerCondition G) {p : ℕ} [Fact p.Prime]
     [Finite (Sylow p G)] (P : Sylow p G) : (↑P : Subgroup G).Normal :=
   normalizer_eq_top.mp <|
     normalizerCondition_iff_only_full_group_self_normalizing.mp hnc _ <| normalizer_normalizer _
 #align sylow.normal_of_normalizer_condition Sylow.normal_of_normalizerCondition
+-/
 
 open scoped BigOperators

bump to nightly-2023-05-28-18

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

8d353152

Diff

@@ -211,7 +211,7 @@ noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p
 instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
   cases nonempty_fintype (Sylow p G); infer_instance
 
-open Pointwise
+open scoped Pointwise
 
 /-- `subgroup.pointwise_mul_action` preserves Sylow subgroups. -/
 instance Sylow.pointwiseMulAction {α : Type _} [Group α] [MulDistribMulAction α G] :
@@ -492,7 +492,7 @@ end InfiniteSylow
 
 open Equiv Equiv.Perm Finset Function List QuotientGroup
 
-open BigOperators
+open scoped BigOperators
 
 universe u v w
 
@@ -715,7 +715,7 @@ theorem subsingleton_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P
 
 section Pointwise
 
-open Pointwise
+open scoped Pointwise
 
 theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
     (h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic :=
@@ -764,7 +764,7 @@ theorem normal_of_normalizerCondition (hnc : NormalizerCondition G) {p : ℕ} [F
     normalizerCondition_iff_only_full_group_self_normalizing.mp hnc _ <| normalizer_normalizer _
 #align sylow.normal_of_normalizer_condition Sylow.normal_of_normalizerCondition
 
-open BigOperators
+open scoped BigOperators
 
 /-- If all its sylow groups are normal, then a finite group is isomorphic to the direct product
 of these sylow groups.

bump to nightly-2023-05-28-04

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

6797a637

Diff

@@ -208,10 +208,8 @@ noncomputable instance (H : Subgroup G) [Fintype (Sylow p G)] : Fintype (Sylow p
   Sylow.fintypeOfInjective H.subtype_injective
 
 /-- If `H` is a subgroup of `G`, then `finite (sylow p G)` implies `finite (sylow p H)`. -/
-instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) :=
-  by
-  cases nonempty_fintype (Sylow p G)
-  infer_instance
+instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
+  cases nonempty_fintype (Sylow p G); infer_instance
 
 open Pointwise
 
@@ -337,10 +335,7 @@ theorem card_sylow_modEq_one [Fact p.Prime] [Fintype (Sylow p G)] : card (Sylow
         _ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩
         _ ↔ Q ∈ {P} := sylow.ext_iff.symm.trans set.mem_singleton_iff.symm
         
-  have : Fintype (fixed_points P.1 (Sylow p G)) :=
-    by
-    rw [this]
-    infer_instance
+  have : Fintype (fixed_points P.1 (Sylow p G)) := by rw [this]; infer_instance
   have : card (fixed_points P.1 (Sylow p G)) = 1 := by simp [this]
   exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])
 #align card_sylow_modeq_one card_sylow_modEq_one
@@ -521,10 +516,7 @@ theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Fin
     inv_mem_iff.1
       (mem_normalizer_fintype fun n (hn : n ∈ H) =>
         have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha (mem_orbit _ ⟨n⁻¹, H.inv_mem hn⟩))
-        show _ ∈ H by
-          rw [mul_inv_rev, inv_inv] at this
-          convert this
-          rw [inv_inv]),
+        show _ ∈ H by rw [mul_inv_rev, inv_inv] at this; convert this; rw [inv_inv]),
     fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
     (mem_fixedPoints' _).2 fun y =>
       Quotient.inductionOn' y fun y hy =>
@@ -543,11 +535,7 @@ def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)]
       normalizer H ⧸ Subgroup.comap ((normalizer H).Subtype : normalizer H →* G) H :=
   @subtypeQuotientEquivQuotientSubtype G (normalizer H : Set G) (id _) (id _) (fixedPoints _ _)
     (fun a => (@mem_fixedPoints_mul_left_cosets_iff_mem_normalizer _ _ _ ‹_› _).symm)
-    (by
-      intros
-      rw [setoidHasEquiv]
-      simp only [left_rel_apply]
-      rfl)
+    (by intros ; rw [setoidHasEquiv]; simp only [left_rel_apply]; rfl)
 #align sylow.fixed_points_mul_left_cosets_equiv_quotient Sylow.fixedPointsMulLeftCosetsEquivQuotient
 
 /-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent

bump to nightly-2023-04-24-06

mathlib commit https://github.com/leanprover-community/mathlib/commit/09079525fd01b3dda35e96adaa08d2f943e1648c

ff662d55

Diff

@@ -503,7 +503,7 @@ universe u v w
 
 variable {G : Type u} {α : Type v} {β : Type w} [Group G]
 
-attribute [local instance] Subtype.fintype setFintype Classical.propDecidable
+attribute [local instance 10] Subtype.fintype setFintype Classical.propDecidable
 
 theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
     Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by

bump to nightly-2023-03-17-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/da3fc4a33ff6bc75f077f691dc94c217b8d41559

624c906c

Diff

@@ -357,7 +357,7 @@ variable {p} {G}
 
 /-- Sylow subgroups are isomorphic -/
 def Sylow.equivSmul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
-  equivSmul (MulAut.conj g) ↑P
+  equivSMul (MulAut.conj g) ↑P
 #align sylow.equiv_smul Sylow.equivSmul
 
 /-- Sylow subgroups are isomorphic -/

bump to nightly-2023-03-01-20

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

20cadee0

Diff

@@ -408,7 +408,7 @@ noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p
   calc
     Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
     _ ≃ orbit G P := by rw [P.orbit_eq_top]
-    _ ≃ G ⧸ stabilizer G P := orbitEquivQuotientStabilizer G P
+    _ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
     _ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
     
 #align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer
@@ -827,7 +827,7 @@ noncomputable def directProductOfNormal [Fintype G]
         congr 1 with ⟨p, hp⟩
         exact @card_eq_multiplicity _ _ _ p ⟨Nat.prime_of_mem_factorization hp⟩ (P p)
       _ = ∏ p in ps, p ^ (card G).factorization p :=
-        Finset.prod_finset_coe (fun p => p ^ (card G).factorization p) _
+        (Finset.prod_finset_coe (fun p => p ^ (card G).factorization p) _)
       _ = (card G).factorization.Prod pow := rfl
       _ = card G := Nat.factorization_prod_pow_eq_self Fintype.card_ne_zero

bump to nightly-2023-02-22-21

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

0652cf4a

Diff

@@ -633,7 +633,8 @@ theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact
     rw [Set.card_image_of_injective
         (Subgroup.comap (mk' (H.subgroup_of H.normalizer)) (zpowers x) : Set H.normalizer)
         Subtype.val_injective,
-      pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, order_eq_card_zpowers, ← Fintype.card_prod]
+      pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, orderOf_eq_card_zpowers, ←
+      Fintype.card_prod]
     exact @Fintype.card_congr _ _ (id _) (id _) (preimage_mk_equiv_subgroup_times_set _ _),
     by
     intro y hy

bump to nightly-2023-02-21-16

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

1107b979

Changes in mathlib4

mathlib3

mathlib4

chore: superfluous parentheses part 2 (#12131)

Co-authored-by: Moritz Firsching <firsching@google.com>

d48d30ac

Diff

@@ -400,7 +400,7 @@ noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Finite (Sylow p
   calc
     Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
     _ ≃ orbit G P := Equiv.setCongr P.orbit_eq_top.symm
-    _ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
+    _ ≃ G ⧸ stabilizer G P := orbitEquivQuotientStabilizer G P
     _ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
 
 #align sylow.equiv_quotient_normalizer Sylow.equivQuotientNormalizer

change the order of operation in zsmulRec and nsmulRec (#11451)

We change the following field in the definition of an additive commutative monoid:

 nsmul_succ : ∀ (n : ℕ) (x : G),
-  AddMonoid.nsmul (n + 1) x = x + AddMonoid.nsmul n x
+  AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x

where the latter is more natural

We adjust the definitions of ^ in monoids, groups, etc. Originally there was a warning comment about why this natural order was preferred

use x * npowRec n x and not npowRec n x * x in the definition to make sure that definitional unfolding of npowRec is blocked, to avoid deep recursion issues.

but it seems to no longer apply.

Remarks on the PR :

pow_succ and pow_succ' have switched their meanings.
Most of the time, the proofs were adjusted by priming/unpriming one lemma, or exchanging left and right; a few proofs were more complicated to adjust.
In particular, [Mathlib/NumberTheory/RamificationInertia.lean] used Ideal.IsPrime.mul_mem_pow which is defined in [Mathlib/RingTheory/DedekindDomain/Ideal.lean]. Changing the order of operation forced me to add the symmetric lemma Ideal.IsPrime.mem_pow_mul.
the docstring for Cauchy condensation test in [Mathlib/Analysis/PSeries.lean] was mathematically incorrect, I added the mention that the function is antitone.

9a3feeae

Diff

@@ -555,7 +555,7 @@ theorem card_normalizer_modEq_card [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact p.
   have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
   rw [card_eq_card_quotient_mul_card_subgroup H,
     card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Fintype.card_congr this,
-    hH, pow_succ]
+    hH, pow_succ']
   exact (card_quotient_normalizer_modEq_card_quotient hH).mul_right' _
 #align sylow.card_normalizer_modeq_card Sylow.card_normalizer_modEq_card
 
@@ -613,7 +613,7 @@ theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact
     rw [Set.card_image_of_injective
         (Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
         Subtype.val_injective,
-      pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
+      pow_succ, ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
       Fintype.card_prod]
     exact @Fintype.card_congr _ _ (_) (_)
       (preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by
@@ -685,7 +685,7 @@ lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {
     exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
   obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
   refine ⟨H', H'H, ?_⟩
-  simpa only [pow_succ, H'card] using And.intro hmk hkm
+  simpa only [pow_succ', H'card] using And.intro hmk hkm
 
 theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
     (hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=

style: homogenise porting notes (#11145)

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

converts "--porting note" into "-- Porting note";
converts "porting note" into "Porting note".

8be0b69c

Diff

@@ -62,7 +62,7 @@ namespace Sylow
 
 attribute [coe] Sylow.toSubgroup
 
---Porting note: Changed to `CoeOut`
+-- Porting note: Changed to `CoeOut`
 instance : CoeOut (Sylow p G) (Subgroup G) :=
   ⟨Sylow.toSubgroup⟩

chore(OrderOfElement,Sylow): Fintype -> Finite (#10550)

aa79ce07

Diff

@@ -395,11 +395,11 @@ theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)]
 #align sylow.conj_eq_normalizer_conj_of_mem Sylow.conj_eq_normalizer_conj_of_mem
 
 /-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
-noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Fintype (Sylow p G)]
+noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Finite (Sylow p G)]
     (P : Sylow p G) : Sylow p G ≃ G ⧸ (P : Subgroup G).normalizer :=
   calc
     Sylow p G ≃ (⊤ : Set (Sylow p G)) := (Equiv.Set.univ (Sylow p G)).symm
-    _ ≃ orbit G P := by rw [P.orbit_eq_top]
+    _ ≃ orbit G P := Equiv.setCongr P.orbit_eq_top.symm
     _ ≃ G ⧸ stabilizer G P := (orbitEquivQuotientStabilizer G P)
     _ ≃ G ⧸ (P : Subgroup G).normalizer := by rw [P.stabilizer_eq_normalizer]
 
@@ -854,7 +854,6 @@ noncomputable def directProductOfNormal [Fintype G]
         (Finset.prod_finset_coe (fun p => p ^ (card G).factorization p) _)
       _ = (card G).factorization.prod (· ^ ·) := rfl
       _ = card G := Nat.factorization_prod_pow_eq_self Fintype.card_ne_zero
-
 #align sylow.direct_product_of_normal Sylow.directProductOfNormal
 
 end Sylow

chore(Order/*): move SupSet, Set.sUnion etc to a new file (#10232)

d18f1d0b

Diff

@@ -9,6 +9,7 @@ import Mathlib.GroupTheory.GroupAction.ConjAct
 import Mathlib.GroupTheory.PGroup
 import Mathlib.GroupTheory.NoncommPiCoprod
 import Mathlib.Order.Atoms.Finite
+import Mathlib.Data.Set.Lattice
 
 #align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"

chore(*): replace $ with <| (#9319)

See Zulip thread for the discussion.

9f6d3388

Diff

@@ -658,13 +658,13 @@ then there is a subgroup of cardinality `p ^ n`. -/
 lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
     (hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
   have : Fact p.Prime := ⟨hp⟩
-  have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
+  have : Finite G := Nat.finite_of_card_ne_zero <| by linarith [Nat.one_le_pow n p hp.pos]
   cases nonempty_fintype G
   obtain ⟨m, hm⟩ := h.exists_card_eq
   simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
   refine exists_subgroup_card_pow_prime _ ?_
   rw [hm] at hn ⊢
-  exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
+  exact pow_dvd_pow _ <| (pow_le_pow_iff_right hp.one_lt).1 hn
 
 /-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
 least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/

doc: Explain relative Sylow theorems (#9158)

Requested by @urkud

e4dff73f

Diff

@@ -653,25 +653,31 @@ theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.P
   ⟨K, hK.1⟩
 #align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
 
-lemma exists_subgroup_card_pow_prime_of_le_card {m p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
-    (hm : p ^ m ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ m := by
+/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group of size at least `p ^ n`
+then there is a subgroup of cardinality `p ^ n`. -/
+lemma exists_subgroup_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
+    (hn : p ^ n ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ n := by
   have : Fact p.Prime := ⟨hp⟩
-  have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow m p hp.pos]
+  have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow n p hp.pos]
   cases nonempty_fintype G
-  obtain ⟨n, hn⟩ := h.exists_card_eq
-  simp_rw [Nat.card_eq_fintype_card] at hn hm ⊢
+  obtain ⟨m, hm⟩ := h.exists_card_eq
+  simp_rw [Nat.card_eq_fintype_card] at hm hn ⊢
   refine exists_subgroup_card_pow_prime _ ?_
-  rw [hn] at hm ⊢
-  exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hm
-
-lemma exists_subgroup_le_card_pow_prime_of_le_card {m p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
-    {H : Subgroup G} (hm : p ^ m ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ m := by
-  obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hm
+  rw [hm] at hn ⊢
+  exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hn
+
+/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
+least `p ^ n` then there is a subgroup of `H` of cardinality `p ^ n`. -/
+lemma exists_subgroup_le_card_pow_prime_of_le_card {n p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
+    {H : Subgroup G} (hn : p ^ n ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ n := by
+  obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hn
   refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
   rw [← H'card]
   let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
   exact Nat.card_congr e.symm.toEquiv
 
+/-- A special case of **Sylow's first theorem**. If `G` is a `p`-group and `H` a subgroup of size at
+least `k` then there is a subgroup of `H` of cardinality between `k / p` and `k`. -/
 lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
     (hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
   obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=

feat: Relative Sylow's first theorems (#8944)

Prove variants of Sylow' first theorem relative to a subgroup.

From PFR

Co-authored-by: Sébastien Gouëzel <sebastien.gouezel@univ-rennes1.fr>

363e9f6e

Diff

@@ -653,6 +653,33 @@ theorem exists_subgroup_card_pow_prime [Fintype G] (p : ℕ) {n : ℕ} [Fact p.P
   ⟨K, hK.1⟩
 #align sylow.exists_subgroup_card_pow_prime Sylow.exists_subgroup_card_pow_prime
 
+lemma exists_subgroup_card_pow_prime_of_le_card {m p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
+    (hm : p ^ m ≤ Nat.card G) : ∃ H : Subgroup G, Nat.card H = p ^ m := by
+  have : Fact p.Prime := ⟨hp⟩
+  have : Finite G := Nat.finite_of_card_ne_zero $ by linarith [Nat.one_le_pow m p hp.pos]
+  cases nonempty_fintype G
+  obtain ⟨n, hn⟩ := h.exists_card_eq
+  simp_rw [Nat.card_eq_fintype_card] at hn hm ⊢
+  refine exists_subgroup_card_pow_prime _ ?_
+  rw [hn] at hm ⊢
+  exact pow_dvd_pow _ $ (pow_le_pow_iff_right hp.one_lt).1 hm
+
+lemma exists_subgroup_le_card_pow_prime_of_le_card {m p : ℕ} (hp : p.Prime) (h : IsPGroup p G)
+    {H : Subgroup G} (hm : p ^ m ≤ Nat.card H) : ∃ H' ≤ H, Nat.card H' = p ^ m := by
+  obtain ⟨H', H'card⟩ := exists_subgroup_card_pow_prime_of_le_card hp (h.to_subgroup H) hm
+  refine ⟨H'.map H.subtype, map_subtype_le _, ?_⟩
+  rw [← H'card]
+  let e : H' ≃* H'.map H.subtype := H'.equivMapOfInjective (Subgroup.subtype H) H.subtype_injective
+  exact Nat.card_congr e.symm.toEquiv
+
+lemma exists_subgroup_le_card_le {k p : ℕ} (hp : p.Prime) (h : IsPGroup p G) {H : Subgroup G}
+    (hk : k ≤ Nat.card H) (hk₀ : k ≠ 0) : ∃ H' ≤ H, Nat.card H' ≤ k ∧ k < p * Nat.card H' := by
+  obtain ⟨m, hmk, hkm⟩ : ∃ s, p ^ s ≤ k ∧ k < p ^ (s + 1) :=
+    exists_nat_pow_near (Nat.one_le_iff_ne_zero.2 hk₀) hp.one_lt
+  obtain ⟨H', H'H, H'card⟩ := exists_subgroup_le_card_pow_prime_of_le_card hp h (hmk.trans hk)
+  refine ⟨H', H'H, ?_⟩
+  simpa only [pow_succ, H'card] using And.intro hmk hkm
+
 theorem pow_dvd_card_of_pow_dvd_card [Fintype G] {p n : ℕ} [hp : Fact p.Prime] (P : Sylow p G)
     (hdvd : p ^ n ∣ card G) : p ^ n ∣ card P :=
   (hp.1.coprime_pow_of_not_dvd

chore: space after ← (#8178)

Co-authored-by: Moritz Firsching <firsching@google.com>

5fb9beab

Diff

@@ -612,7 +612,7 @@ theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact
     rw [Set.card_image_of_injective
         (Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
         Subtype.val_injective,
-      pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ←Fintype.card_zpowers, ←
+      pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ← Fintype.card_zpowers, ←
       Fintype.card_prod]
     exact @Fintype.card_congr _ _ (_) (_)
       (preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by

feat: Order of elements of a subgroup (#8385)

The cardinality of a subgroup is greater than the order of any of its elements.

Rename

order_eq_card_zpowers → Fintype.card_zpowers
order_eq_card_zpowers' → Nat.card_zpowers (and turn it around to match Nat.card_subgroupPowers)
Submonoid.powers_subset → Submonoid.powers_le
orderOf_dvd_card_univ → orderOf_dvd_card
orderOf_subgroup → Subgroup.orderOf
Subgroup.nonempty → Subgroup.coe_nonempty

ee3bd06d

Diff

@@ -430,7 +430,7 @@ theorem not_dvd_index_sylow' [hp : Fact p.Prime] (P : Sylow p G) [(P : Subgroup
   letI : Fintype (G ⧸ (P : Subgroup G)) := (P : Subgroup G).fintypeQuotientOfFiniteIndex
   rw [index_eq_card (P : Subgroup G)] at h
   obtain ⟨x, hx⟩ := exists_prime_orderOf_dvd_card (G := G ⧸ (P : Subgroup G)) p h
-  have h := IsPGroup.of_card ((orderOf_eq_card_zpowers.symm.trans hx).trans (pow_one p).symm)
+  have h := IsPGroup.of_card ((Fintype.card_zpowers.trans hx).trans (pow_one p).symm)
   let Q := (zpowers x).comap (QuotientGroup.mk' (P : Subgroup G))
   have hQ : IsPGroup p Q := by
     apply h.comap_of_ker_isPGroup
@@ -612,7 +612,7 @@ theorem exists_subgroup_card_pow_succ [Fintype G] {p : ℕ} {n : ℕ} [hp : Fact
     rw [Set.card_image_of_injective
         (Subgroup.comap (mk' (H.subgroupOf H.normalizer)) (zpowers x) : Set H.normalizer)
         Subtype.val_injective,
-      pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, orderOf_eq_card_zpowers, ←
+      pow_succ', ← hH, Fintype.card_congr hequiv, ← hx, ←Fintype.card_zpowers, ←
       Fintype.card_prod]
     exact @Fintype.card_congr _ _ (_) (_)
       (preimageMkEquivSubgroupProdSet (H.subgroupOf H.normalizer) (zpowers x)), by

https://github.com/leanprover-community/mathlib4/blob/96a11c7aac574c00370c2b3dab483cb676405c5d/Mathlib/Logic/Unique.lean#L255-L256

fix: attribute [simp] ... in -> attribute [local simp] ... in (#7678)

Mathlib.Logic.Unique contains the line attribute [simp] eq_iff_true_of_subsingleton in ...:

Despite what the in part may imply, this adds the lemma to the simp set "globally", including for downstream files; it is likely that attribute [local simp] eq_iff_true_of_subsingleton in ... was meant instead (or maybe scoped simp, but I think "scoped" refers to the current namespace). Indeed, the relevant lemma is not marked with @[simp] for possible slowness: https://github.com/leanprover/std4/blob/846e9e1d6bb534774d1acd2dc430e70987da3c18/Std/Logic.lean#L749. Adding it to the simp set causes the example at https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Regression.20in.20simp to slow down.

This PR changes this and fixes the relevant downstream simps. There was also one ocurrence of attribute [simp] FullSubcategory.comp_def FullSubcategory.id_def in in Mathlib.CategoryTheory.Monoidal.Subcategory but that was much easier to fix.

https://github.com/leanprover-community/mathlib4/blob/bc49eb9ba756a233370b4b68bcdedd60402f71ed/Mathlib/CategoryTheory/Monoidal/Subcategory.lean#L118-L119

1fe87718

Diff

@@ -731,7 +731,7 @@ theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (
   intro Φ
   show (Φ • P).toSubgroup = P.toSubgroup
   congr
-  simp
+  simp [eq_iff_true_of_subsingleton]
 #align sylow.characteristic_of_normal Sylow.characteristic_of_normal
 
 end Pointwise

feat: measurability of uniqueElim and piFinsetUnion (#8249)

The extra import doesn't add any additional imports transitively
Also use implicit arguments for some MeasurableSpace arguments for consistency with other lemmas.
Redefine Equiv.piSubsingleton and MulEquiv.piSubsingleton to Equiv.piUnique and MulEquiv.piUnique.

76efc30c

Diff

@@ -710,15 +710,15 @@ theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintyp
   rfl
 #align sylow.coe_of_card Sylow.coe_ofCard
 
-theorem subsingleton_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
-    (h : (P : Subgroup G).Normal) : Subsingleton (Sylow p G) := by
-  apply Subsingleton.intro
-  intro Q R
+/-- If `G` has a normal Sylow `p`-subgroup, then it is the only Sylow `p`-subgroup. -/
+noncomputable def unique_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
+    (h : (P : Subgroup G).Normal) : Unique (Sylow p G) := by
+  refine { uniq := fun Q ↦ ?_ }
   obtain ⟨x, h1⟩ := exists_smul_eq G P Q
-  obtain ⟨x, h2⟩ := exists_smul_eq G P R
+  obtain ⟨x, h2⟩ := exists_smul_eq G P default
   rw [Sylow.smul_eq_of_normal] at h1 h2
   rw [← h1, ← h2]
-#align sylow.subsingleton_of_normal Sylow.subsingleton_of_normal
+#align sylow.subsingleton_of_normal Sylow.unique_of_normal
 
 section Pointwise
 
@@ -726,7 +726,7 @@ open Pointwise
 
 theorem characteristic_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
     (h : (P : Subgroup G).Normal) : (P : Subgroup G).Characteristic := by
-  haveI := Sylow.subsingleton_of_normal P h
+  haveI := Sylow.unique_of_normal P h
   rw [characteristic_iff_map_eq]
   intro Φ
   show (Φ • P).toSubgroup = P.toSubgroup
@@ -796,9 +796,8 @@ noncomputable def directProductOfNormal [Fintype G]
     apply @MulEquiv.piCongrRight ps (fun p => ∀ P : Sylow p G, P) (fun p => P p) _ _
     rintro ⟨p, hp⟩
     haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
-    haveI := subsingleton_of_normal _ (hn (P p))
-    change (∀ P : Sylow p G, P) ≃* P p
-    exact MulEquiv.piSubsingleton _ _
+    letI := unique_of_normal _ (hn (P p))
+    apply MulEquiv.piUnique
   show (∀ p : ps, P p) ≃* G
   apply MulEquiv.ofBijective (Subgroup.noncommPiCoprod hcomm)
   apply (bijective_iff_injective_and_card _).mpr

refactor: Unify spelling of "prime factors" (#8164)

mathlib can't make up its mind on whether to spell "the prime factors of n" as n.factors.toFinset or n.factorization.support, even though those two are defeq. This PR proposes to unify everything to a new definition Nat.primeFactors, and streamline the existing scattered API about n.factors.toFinset and n.factorization.support to Nat.primeFactors. We also get to write a bit more API that didn't make sense before, eg primeFactors_mono.

5ced22e4

Diff

@@ -778,14 +778,14 @@ of these Sylow subgroups.
 -/
 noncomputable def directProductOfNormal [Fintype G]
     (hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
-    (∀ p : (card G).factorization.support, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
-  set ps := (Fintype.card G).factorization.support
+    (∀ p : (card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
+  set ps := (Fintype.card G).primeFactors
   -- “The” Sylow subgroup for p
   let P : ∀ p, Sylow p G := default
   have hcomm : Pairwise fun p₁ p₂ : ps => ∀ x y : G, x ∈ P p₁ → y ∈ P p₂ → Commute x y := by
     rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne
-    haveI hp₁' := Fact.mk (Nat.prime_of_mem_factorization hp₁)
-    haveI hp₂' := Fact.mk (Nat.prime_of_mem_factorization hp₂)
+    haveI hp₁' := Fact.mk (Nat.prime_of_mem_primeFactors hp₁)
+    haveI hp₂' := Fact.mk (Nat.prime_of_mem_primeFactors hp₂)
     have hne' : p₁ ≠ p₂ := by simpa using hne
     apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P p₁)) (hn (P p₂))
     apply IsPGroup.disjoint_of_ne p₁ p₂ hne' _ _ (P p₁).isPGroup' (P p₂).isPGroup'
@@ -795,7 +795,7 @@ noncomputable def directProductOfNormal [Fintype G]
   · -- here we need to help the elaborator with an explicit instantiation
     apply @MulEquiv.piCongrRight ps (fun p => ∀ P : Sylow p G, P) (fun p => P p) _ _
     rintro ⟨p, hp⟩
-    haveI hp' := Fact.mk (Nat.prime_of_mem_factorization hp)
+    haveI hp' := Fact.mk (Nat.prime_of_mem_primeFactors hp)
     haveI := subsingleton_of_normal _ (hn (P p))
     change (∀ P : Sylow p G, P) ≃* P p
     exact MulEquiv.piSubsingleton _ _
@@ -807,8 +807,8 @@ noncomputable def directProductOfNormal [Fintype G]
   · apply Subgroup.injective_noncommPiCoprod_of_independent
     apply independent_of_coprime_order hcomm
     rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne
-    haveI hp₁' := Fact.mk (Nat.prime_of_mem_factorization hp₁)
-    haveI hp₂' := Fact.mk (Nat.prime_of_mem_factorization hp₂)
+    haveI hp₁' := Fact.mk (Nat.prime_of_mem_primeFactors hp₁)
+    haveI hp₂' := Fact.mk (Nat.prime_of_mem_primeFactors hp₂)
     have hne' : p₁ ≠ p₂ := by simpa using hne
     apply IsPGroup.coprime_card_of_ne p₁ p₂ hne' _ _ (P p₁).isPGroup' (P p₂).isPGroup'
   show card (∀ p : ps, P p) = card G
@@ -816,7 +816,7 @@ noncomputable def directProductOfNormal [Fintype G]
       card (∀ p : ps, P p) = ∏ p : ps, card (P p) := Fintype.card_pi
       _ = ∏ p : ps, p.1 ^ (card G).factorization p.1 := by
         congr 1 with ⟨p, hp⟩
-        exact @card_eq_multiplicity _ _ _ p ⟨Nat.prime_of_mem_factorization hp⟩ (P p)
+        exact @card_eq_multiplicity _ _ _ p ⟨Nat.prime_of_mem_primeFactors hp⟩ (P p)
       _ = ∏ p in ps, p ^ (card G).factorization p :=
         (Finset.prod_finset_coe (fun p => p ^ (card G).factorization p) _)
       _ = (card G).factorization.prod (· ^ ·) := rfl

refactor(GroupTheory/GroupAction/Basic): re-organise, rename, and make some variables implicit (#7786)

Re-organise the namespace and section structure of GroupTheory/GroupAction/Basic.lean.
Remove the namespaces MulAction.Stabilizer and AddAction.Stabilizer, renaming MulAction.Stabilizer.submonoid to MulAction.stabilizerSubmonoid.
Make variables for the monoid/group/set implicit when an element or subset is used in the statement.

51dd9ef2

Diff

@@ -503,7 +503,7 @@ namespace Sylow
 theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
     {x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
   ⟨fun hx =>
-    have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := (mem_fixedPoints' _).1 hx _
+    have ha : ∀ {y : G ⧸ H}, y ∈ orbit H (x : G ⧸ H) → y = x := mem_fixedPoints'.1 hx _
     (inv_mem_iff (G := G)).1
       (mem_normalizer_fintype fun n (hn : n ∈ H) =>
         have : (n⁻¹ * x)⁻¹ * x ∈ H := QuotientGroup.eq.1 (ha ⟨⟨n⁻¹, inv_mem hn⟩, rfl⟩)
@@ -512,7 +512,7 @@ theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Fin
           convert this
           rw [inv_inv]),
     fun hx : ∀ n : G, n ∈ H ↔ x * n * x⁻¹ ∈ H =>
-    (mem_fixedPoints' _).2 fun y =>
+    mem_fixedPoints'.2 fun y =>
       Quotient.inductionOn' y fun y hy =>
         QuotientGroup.eq.2
           (let ⟨⟨b, hb₁⟩, hb₂⟩ := hy

feat: small missing group lemmas (#7614)

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

644ffced

Diff

@@ -500,9 +500,6 @@ theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G
 #align quotient_group.card_preimage_mk QuotientGroup.card_preimage_mk
 
 namespace Sylow
-
-open Subgroup Submonoid MulAction
-
 theorem mem_fixedPoints_mul_left_cosets_iff_mem_normalizer {H : Subgroup G} [Finite (H : Set G)]
     {x : G} : (x : G ⧸ H) ∈ MulAction.fixedPoints H (G ⧸ H) ↔ x ∈ normalizer H :=
   ⟨fun hx =>

chore: bump to v4.1.0-rc1 (2nd attempt) (#7216)

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

bfaffcf1

Diff

@@ -672,7 +672,7 @@ theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G
 
 /-- Sylow subgroups are Hall subgroups. -/
 theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
-    (card P).coprime (index (P : Subgroup G)) :=
+    (card P).Coprime (index (P : Subgroup G)) :=
   let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
   hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
 #align sylow.card_coprime_index Sylow.card_coprime_index

Revert "chore: bump to v4.1.0-rc1 (#7174)" (#7198)

This reverts commit 6f8e8104. Unfortunately this bump was not linted correctly, as CI did not run runLinter Mathlib.

We can unrevert once that's fixed.

40b58304

Diff

@@ -672,7 +672,7 @@ theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G
 
 /-- Sylow subgroups are Hall subgroups. -/
 theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
-    (card P).Coprime (index (P : Subgroup G)) :=
+    (card P).coprime (index (P : Subgroup G)) :=
   let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
   hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
 #align sylow.card_coprime_index Sylow.card_coprime_index

chore: bump to v4.1.0-rc1 (#7174)

Some changes have already been review and delegated in #6910 and #7148.

The diff that needs looking at is https://github.com/leanprover-community/mathlib4/pull/7174/commits/64d6d07ee18163627c8f517eb31455411921c5ac

The std bump PR was insta-merged already!

Co-authored-by: leanprover-community-mathlib4-bot <leanprover-community-mathlib4-bot@users.noreply.github.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

6f8e8104

Diff

@@ -672,7 +672,7 @@ theorem dvd_card_of_dvd_card [Fintype G] {p : ℕ} [Fact p.Prime] (P : Sylow p G
 
 /-- Sylow subgroups are Hall subgroups. -/
 theorem card_coprime_index [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
-    (card P).coprime (index (P : Subgroup G)) :=
+    (card P).Coprime (index (P : Subgroup G)) :=
   let ⟨_n, hn⟩ := IsPGroup.iff_card.mp P.2
   hn.symm ▸ (hp.1.coprime_pow_of_not_dvd (not_dvd_index_sylow P index_ne_zero_of_finite)).symm
 #align sylow.card_coprime_index Sylow.card_coprime_index

refactor(*): Protect Function.Commute (#6456)

This PR protects Function.Commute, so that it no longer clashes with Commute in the root namespace, as suggested by @j-loreaux in #6290.

abbba041

Diff

@@ -785,7 +785,7 @@ noncomputable def directProductOfNormal [Fintype G]
   set ps := (Fintype.card G).factorization.support
   -- “The” Sylow subgroup for p
   let P : ∀ p, Sylow p G := default
-  have hcomm : Pairwise fun p₁ p₂ : ps => ∀ x y : G, x ∈ P p₁ → y ∈ P p₂ → _root_.Commute x y := by
+  have hcomm : Pairwise fun p₁ p₂ : ps => ∀ x y : G, x ∈ P p₁ → y ∈ P p₂ → Commute x y := by
     rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne
     haveI hp₁' := Fact.mk (Nat.prime_of_mem_factorization hp₁)
     haveI hp₂' := Fact.mk (Nat.prime_of_mem_factorization hp₂)

chore: banish Type _ and Sort _ (#6499)

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

32de3f67

Diff

@@ -47,7 +47,7 @@ open Fintype MulAction Subgroup
 
 section InfiniteSylow
 
-variable (p : ℕ) (G : Type _) [Group G]
+variable (p : ℕ) (G : Type*) [Group G]
 
 /-- A Sylow `p`-subgroup is a maximal `p`-subgroup. -/
 structure Sylow extends Subgroup G where
@@ -91,11 +91,11 @@ instance : SubgroupClass (Sylow p G) G where
 variable (P : Sylow p G)
 
 /-- The action by a Sylow subgroup is the action by the underlying group. -/
-instance mulActionLeft {α : Type _} [MulAction G α] : MulAction P α :=
+instance mulActionLeft {α : Type*} [MulAction G α] : MulAction P α :=
   inferInstanceAs (MulAction (P : Subgroup G) α)
 #align sylow.mul_action_left Sylow.mulActionLeft
 
-variable {K : Type _} [Group K] (ϕ : K →* G) {N : Subgroup G}
+variable {K : Type*} [Group K] (ϕ : K →* G) {N : Subgroup G}
 
 /-- The preimage of a Sylow subgroup under a p-group-kernel homomorphism is a Sylow subgroup. -/
 def comapOfKerIsPGroup (hϕ : IsPGroup p ϕ.ker) (h : ↑P ≤ ϕ.range) : Sylow p K :=
@@ -169,13 +169,13 @@ noncomputable instance Sylow.inhabited : Inhabited (Sylow p G) :=
   Classical.inhabited_of_nonempty Sylow.nonempty
 #align sylow.inhabited Sylow.inhabited
 
-theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type _} [Group H] (P : Sylow p H) {f : H →* G}
+theorem Sylow.exists_comap_eq_of_ker_isPGroup {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
     (hf : IsPGroup p f.ker) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
   Exists.imp (fun Q hQ => P.3 (Q.2.comap_of_ker_isPGroup f hf) (map_le_iff_le_comap.mp hQ))
     (P.2.map f).exists_le_sylow
 #align sylow.exists_comap_eq_of_ker_is_p_group Sylow.exists_comap_eq_of_ker_isPGroup
 
-theorem Sylow.exists_comap_eq_of_injective {H : Type _} [Group H] (P : Sylow p H) {f : H →* G}
+theorem Sylow.exists_comap_eq_of_injective {H : Type*} [Group H] (P : Sylow p H) {f : H →* G}
     (hf : Function.Injective f) : ∃ Q : Sylow p G, (Q : Subgroup G).comap f = P :=
   P.exists_comap_eq_of_ker_isPGroup (IsPGroup.ker_isPGroup_of_injective hf)
 #align sylow.exists_comap_eq_of_injective Sylow.exists_comap_eq_of_injective
@@ -187,7 +187,7 @@ theorem Sylow.exists_comap_subtype_eq {H : Subgroup G} (P : Sylow p H) :
 
 /-- If the kernel of `f : H →* G` is a `p`-group,
   then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
-noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type _} [Group H] {f : H →* G}
+noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type*} [Group H] {f : H →* G}
     (hf : IsPGroup p f.ker) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
   let h_exists := fun P : Sylow p H => P.exists_comap_eq_of_ker_isPGroup hf
   let g : Sylow p H → Sylow p G := fun P => Classical.choose (h_exists P)
@@ -196,7 +196,7 @@ noncomputable def Sylow.fintypeOfKerIsPGroup {H : Type _} [Group H] {f : H →*
 #align sylow.fintype_of_ker_is_p_group Sylow.fintypeOfKerIsPGroup
 
 /-- If `f : H →* G` is injective, then `Fintype (Sylow p G)` implies `Fintype (Sylow p H)`. -/
-noncomputable def Sylow.fintypeOfInjective {H : Type _} [Group H] {f : H →* G}
+noncomputable def Sylow.fintypeOfInjective {H : Type*} [Group H] {f : H →* G}
     (hf : Function.Injective f) [Fintype (Sylow p G)] : Fintype (Sylow p H) :=
   Sylow.fintypeOfKerIsPGroup (IsPGroup.ker_isPGroup_of_injective hf)
 #align sylow.fintype_of_injective Sylow.fintypeOfInjective
@@ -213,7 +213,7 @@ instance (H : Subgroup G) [Finite (Sylow p G)] : Finite (Sylow p H) := by
 open Pointwise
 
 /-- `Subgroup.pointwiseMulAction` preserves Sylow subgroups. -/
-instance Sylow.pointwiseMulAction {α : Type _} [Group α] [MulDistribMulAction α G] :
+instance Sylow.pointwiseMulAction {α : Type*} [Group α] [MulDistribMulAction α G] :
     MulAction α (Sylow p G) where
   smul g P :=
     ⟨(g • P.toSubgroup : Subgroup G), P.2.map _, fun {Q} hQ hS =>
@@ -225,7 +225,7 @@ instance Sylow.pointwiseMulAction {α : Type _} [Group α] [MulDistribMulAction
   mul_smul g h P := Sylow.ext (mul_smul g h P.toSubgroup)
 #align sylow.pointwise_mul_action Sylow.pointwiseMulAction
 
-theorem Sylow.pointwise_smul_def {α : Type _} [Group α] [MulDistribMulAction α G] {g : α}
+theorem Sylow.pointwise_smul_def {α : Type*} [Group α] [MulDistribMulAction α G] {g : α}
     {P : Sylow p G} : ↑(g • P) = g • (P : Subgroup G) :=
   rfl
 #align sylow.pointwise_smul_def Sylow.pointwise_smul_def

chore(GroupTheory): forward-port leanprover-community/mathlib#18965 (#6147)

13e7c2f0

Diff

@@ -10,7 +10,7 @@ import Mathlib.GroupTheory.PGroup
 import Mathlib.GroupTheory.NoncommPiCoprod
 import Mathlib.Order.Atoms.Finite
 
-#align_import group_theory.sylow from "leanprover-community/mathlib"@"bd365b1a4901dbd878e86cb146c2bd86533df468"
+#align_import group_theory.sylow from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
 
 /-!
 # Sylow theorems
@@ -370,17 +370,18 @@ theorem Sylow.stabilizer_eq_normalizer (P : Sylow p G) :
 #align sylow.stabilizer_eq_normalizer Sylow.stabilizer_eq_normalizer
 
 theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite (Sylow p G)]
-    (P : Sylow p G) (x g : G) (hx : x ∈ (P : Subgroup G).centralizer)
-    (hy : g⁻¹ * x * g ∈ (P : Subgroup G).centralizer) :
+    (P : Sylow p G) (x g : G) (hx : x ∈ centralizer (P : Set G))
+    (hy : g⁻¹ * x * g ∈ centralizer (P : Set G)) :
     ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n := by
-  have h1 : ↑P ≤ (zpowers x).centralizer := by rwa [le_centralizer_iff, zpowers_le]
-  have h2 : ↑(g • P) ≤ (zpowers x).centralizer := by
+  have h1 : ↑P ≤ centralizer (zpowers x : Set G) := by rwa [le_centralizer_iff, zpowers_le]
+  have h2 : ↑(g • P) ≤ centralizer (zpowers x : Set G) := by
     rw [le_centralizer_iff, zpowers_le]
     rintro - ⟨z, hz, rfl⟩
     specialize hy z hz
     rwa [← mul_assoc, ← eq_mul_inv_iff_mul_eq, mul_assoc, mul_assoc, mul_assoc, ← mul_assoc,
       eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc] at hy
-  obtain ⟨h, hh⟩ := exists_smul_eq (zpowers x).centralizer ((g • P).subtype h2) (P.subtype h1)
+  obtain ⟨h, hh⟩ :=
+    exists_smul_eq (centralizer (zpowers x : Set G)) ((g • P).subtype h2) (P.subtype h1)
   simp_rw [Sylow.smul_subtype, Subgroup.smul_def, smul_smul] at hh
   refine' ⟨h * g, Sylow.smul_eq_iff_mem_normalizer.mp (Sylow.subtype_injective hh), _⟩
   rw [← mul_assoc, Commute.right_comm (h.prop x (mem_zpowers x)), mul_inv_rev, inv_mul_cancel_right]

chore: use · instead of . (#6085)

898a8e78

Diff

@@ -821,7 +821,7 @@ noncomputable def directProductOfNormal [Fintype G]
         exact @card_eq_multiplicity _ _ _ p ⟨Nat.prime_of_mem_factorization hp⟩ (P p)
       _ = ∏ p in ps, p ^ (card G).factorization p :=
         (Finset.prod_finset_coe (fun p => p ^ (card G).factorization p) _)
-      _ = (card G).factorization.prod (. ^ .) := rfl
+      _ = (card G).factorization.prod (· ^ ·) := rfl
       _ = card G := Nat.factorization_prod_pow_eq_self Fintype.card_ne_zero
 
 #align sylow.direct_product_of_normal Sylow.directProductOfNormal

chore: script to replace headers with #align_import statements (#5979)

depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

89aa3a3b

Diff

@@ -2,11 +2,6 @@
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Thomas Browning
-
-! This file was ported from Lean 3 source module group_theory.sylow
-! leanprover-community/mathlib commit bd365b1a4901dbd878e86cb146c2bd86533df468
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Data.Nat.Factorization.Basic
 import Mathlib.Data.SetLike.Fintype
@@ -15,6 +10,8 @@ import Mathlib.GroupTheory.PGroup
 import Mathlib.GroupTheory.NoncommPiCoprod
 import Mathlib.Order.Atoms.Finite
 
+#align_import group_theory.sylow from "leanprover-community/mathlib"@"bd365b1a4901dbd878e86cb146c2bd86533df468"
+
 /-!
 # Sylow theorems

feat(GroupTheory/Sylow): add inverse to card_eq_multiplicity (#5831)

Port https://github.com/leanprover-community/mathlib/pull/18300 to mathlib4

350feed7

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Thomas Browning
 
 ! This file was ported from Lean 3 source module group_theory.sylow
-! leanprover-community/mathlib commit c55911f6166b348e1c72b08c8664e3af5f1ce334
+! leanprover-community/mathlib commit bd365b1a4901dbd878e86cb146c2bd86533df468
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -37,7 +37,7 @@ The Sylow theorems are the following results for every finite group `G` and ever
   there exists a subgroup of `G` of order `pⁿ`.
 * `IsPGroup.exists_le_sylow`: A generalization of Sylow's first theorem:
   Every `p`-subgroup is contained in a Sylow `p`-subgroup.
-* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow group is `p ^ n`
+* `Sylow.card_eq_multiplicity`: The cardinality of a Sylow subgroup is `p ^ n`
  where `n` is the multiplicity of `p` in the group order.
 * `sylow_conjugate`: A generalization of Sylow's second theorem:
   If the number of Sylow `p`-subgroups is finite, then all Sylow `p`-subgroups are conjugate.
@@ -686,7 +686,7 @@ theorem ne_bot_of_dvd_card [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow
   rwa [h, card_bot] at key
 #align sylow.ne_bot_of_dvd_card Sylow.ne_bot_of_dvd_card
 
-/-- The cardinality of a Sylow group is `p ^ n`
+/-- The cardinality of a Sylow subgroup is `p ^ n`
  where `n` is the multiplicity of `p` in the group order. -/
 theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylow p G) :
     card P = p ^ Nat.factorization (card G) p := by
@@ -696,6 +696,25 @@ theorem card_eq_multiplicity [Fintype G] {p : ℕ} [hp : Fact p.Prime] (P : Sylo
   exact P.1.card_subgroup_dvd_card
 #align sylow.card_eq_multiplicity Sylow.card_eq_multiplicity
 
+/-- A subgroup with cardinality `p ^ n` is a Sylow subgroup
+ where `n` is the multiplicity of `p` in the group order. -/
+def ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
+    (card_eq : card H = p ^ (card G).factorization p) : Sylow p G
+    where
+  toSubgroup := H
+  isPGroup' := IsPGroup.of_card card_eq
+  is_maximal' := by
+    obtain ⟨P, hHP⟩ := (IsPGroup.of_card card_eq).exists_le_sylow
+    exact SetLike.ext'
+      (Set.eq_of_subset_of_card_le hHP (P.card_eq_multiplicity.trans card_eq.symm).le).symm ▸ P.3
+#align sylow.of_card Sylow.ofCard
+
+@[simp, norm_cast]
+theorem coe_ofCard [Fintype G] {p : ℕ} [Fact p.Prime] (H : Subgroup G) [Fintype H]
+    (card_eq : card H = p ^ (card G).factorization p) : ↑(ofCard H card_eq) = H :=
+  rfl
+#align sylow.coe_of_card Sylow.coe_ofCard
+
 theorem subsingleton_of_normal {p : ℕ} [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
     (h : (P : Subgroup G).Normal) : Subsingleton (Sylow p G) := by
   apply Subsingleton.intro
@@ -759,14 +778,14 @@ theorem normal_of_normalizerCondition (hnc : NormalizerCondition G) {p : ℕ} [F
 
 open BigOperators
 
-/-- If all its sylow groups are normal, then a finite group is isomorphic to the direct product
-of these sylow groups.
+/-- If all its Sylow subgroups are normal, then a finite group is isomorphic to the direct product
+of these Sylow subgroups.
 -/
 noncomputable def directProductOfNormal [Fintype G]
     (hn : ∀ {p : ℕ} [Fact p.Prime] (P : Sylow p G), (↑P : Subgroup G).Normal) :
     (∀ p : (card G).factorization.support, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G := by
   set ps := (Fintype.card G).factorization.support
-  -- “The” sylow group for p
+  -- “The” Sylow subgroup for p
   let P : ∀ p, Sylow p G := default
   have hcomm : Pairwise fun p₁ p₂ : ps => ∀ x y : G, x ∈ P p₁ → y ∈ P p₂ → _root_.Commute x y := by
     rintro ⟨p₁, hp₁⟩ ⟨p₂, hp₂⟩ hne
@@ -776,7 +795,7 @@ noncomputable def directProductOfNormal [Fintype G]
     apply Subgroup.commute_of_normal_of_disjoint _ _ (hn (P p₁)) (hn (P p₂))
     apply IsPGroup.disjoint_of_ne p₁ p₂ hne' _ _ (P p₁).isPGroup' (P p₂).isPGroup'
   refine' MulEquiv.trans (N := ∀ p : ps, P p) _ _
-  -- There is only one sylow group for each p, so the inner product is trivial
+  -- There is only one Sylow subgroup for each p, so the inner product is trivial
   show (∀ p : ps, ∀ P : Sylow p G, P) ≃* ∀ p : ps, P p
   · -- here we need to help the elaborator with an explicit instantiation
     apply @MulEquiv.piCongrRight ps (fun p => ∀ P : Sylow p G, P) (fun p => P p) _ _

chore: clean up spacing around at and goals (#5387)

Changes are of the form

some_tactic at h⊢ -> some_tactic at h ⊢
some_tactic at h -> some_tactic at h

2a870323

Diff

@@ -139,7 +139,7 @@ theorem coe_subtype (h : ↑P ≤ N) : ↑(P.subtype h) = subgroupOf (↑P) N :=
 
 theorem subtype_injective {P Q : Sylow p G} {hP : ↑P ≤ N} {hQ : ↑Q ≤ N}
     (h : P.subtype hP = Q.subtype hQ) : P = Q := by
-  rw [SetLike.ext_iff] at h⊢
+  rw [SetLike.ext_iff] at h ⊢
   exact fun g => ⟨fun hg => (h ⟨g, hP hg⟩).mp hg, fun hg => (h ⟨g, hQ hg⟩).mpr hg⟩
 #align sylow.subtype_injective Sylow.subtype_injective
 
@@ -159,7 +159,7 @@ theorem IsPGroup.exists_le_sylow {P : Subgroup G} (hP : IsPGroup p P) : ∃ Q :
                 ⟨R, ⟨R, rfl⟩, R.1.mul_mem hg (T hh)⟩ },
           fun ⟨g, _, ⟨S, rfl⟩, hg⟩ => by
           refine' Exists.imp (fun k hk => _) (hc1 S.2 ⟨g, hg⟩)
-          rwa [Subtype.ext_iff, coe_pow] at hk⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
+          rwa [Subtype.ext_iff, coe_pow] at hk ⊢, fun M hM g hg => ⟨M, ⟨⟨M, hM⟩, rfl⟩, hg⟩⟩)
       P hP)
     fun {Q} ⟨hQ1, hQ2, hQ3⟩ => ⟨⟨Q, hQ1, hQ3 _⟩, hQ2⟩
 #align is_p_group.exists_le_sylow IsPGroup.exists_le_sylow