l3lab/ntp-mathlib huggingface dataset:
/-- 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}
    (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)
  have 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 

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
H : Type u_2
inst✝¹ : Group H
f : H →* G
hf : IsPGroup p ↥(MonoidHom.ker f)
inst✝ : Fintype (Sylow p G)
h_exists : ∀ (P : Sylow p H), ∃ Q, comap f ↑Q = ↑P := fun P => exists_comap_eq_of_ker_isPGroup P hf
g : Sylow p H → Sylow p G := fun P => Classical.choose (_ : ∃ Q, comap f ↑Q = ↑P)
hg : ∀ (P : Sylow p H), comap f ↑(g P) = ↑P
P Q : Sylow p H
h : g P = g Q
⊢ ↑P = ↑Q

Next tactic:
rw [← hg, 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}
    (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)
  have 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 rw [← hg, h]; 

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
H : Type u_2
inst✝¹ : Group H
f : H →* G
hf : IsPGroup p ↥(MonoidHom.ker f)
inst✝ : Fintype (Sylow p G)
h_exists : ∀ (P : Sylow p H), ∃ Q, comap f ↑Q = ↑P := fun P => exists_comap_eq_of_ker_isPGroup P hf
g : Sylow p H → Sylow p G := fun P => Classical.choose (_ : ∃ Q, comap f ↑Q = ↑P)
hg : ∀ (P : Sylow p H), comap f ↑(g P) = ↑P
P Q : Sylow p H
h : g P = g Q
⊢ comap f ↑(g Q) = ↑Q

Next tactic:
exact (h_exists Q).choose_spec
/-- 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
  

Proof state:
p : ℕ
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite (Sylow p G)
⊢ Finite (Sylow p ↥H)

Next tactic:
cases nonempty_fintype (Sylow p G)
/-- 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)
  

Proof state:
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
inst✝ : Finite (Sylow p G)
val✝ : Fintype (Sylow p G)
⊢ Finite (Sylow p ↥H)

Next tactic:
infer_instance
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
    g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
  

Proof state:
p : ℕ
G : Type u_1
inst✝ : Group G
g : G
P : Sylow p G
⊢ g • P = P ↔ g ∈ normalizer ↑P

Next tactic:
rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
      mem_normalizer_iff, inv_inv]
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
    g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
  rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
      mem_normalizer_iff, inv_inv]
  

Proof state:
p : ℕ
G : Type u_1
inst✝ : Group G
g : G
P : Sylow p G
⊢ (∀ (x : G), x ∈ P ↔ x ∈ g • P) ↔ ∀ (h : G), h ∈ ↑P ↔ g⁻¹ * h * g ∈ ↑P

Next tactic:
exact
    forall_congr' fun h =>
      iff_congr Iff.rfl
        ⟨fun ⟨a, b, c⟩ => c ▸ by simpa [mul_assoc] using b,
          fun hh => ⟨(MulAut.conj g)⁻¹ h, hh, MulAut.apply_inv_self G (MulAut.conj g) h⟩⟩
theorem Sylow.smul_eq_iff_mem_normalizer {g : G} {P : Sylow p G} :
    g • P = P ↔ g ∈ (P : Subgroup G).normalizer := by
  rw [eq_comm, SetLike.ext_iff, ← inv_mem_iff (G := G) (H := normalizer P.toSubgroup),
      mem_normalizer_iff, inv_inv]
  exact
    forall_congr' fun h =>
      iff_congr Iff.rfl
        ⟨fun ⟨a, b, c⟩ => c ▸ by 

Proof state:
p : ℕ
G : Type u_1
inst✝ : Group G
g : G
P : Sylow p G
h : G
x✝ : h ∈ g • P
a : G
b : a ∈ ↑↑P
c : ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) a = h
⊢ g⁻¹ * ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) a * g ∈ ↑P

Next tactic:
simpa [mul_assoc] using b
theorem Sylow.smul_eq_of_normal {g : G} {P : Sylow p G} [h : (P : Subgroup G).Normal] : g • P = P :=
  by 

Proof state:
p : ℕ
G : Type u_1
inst✝ : Group G
g : G
P : Sylow p G
h : Normal ↑P
⊢ g • P = P

Next tactic:
simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top]
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
  

Proof state:
p : ℕ
G : Type u_1
inst✝ : Group G
H : Subgroup G
P : Sylow p G
⊢ P ∈ fixedPoints (↥H) (Sylow p G) ↔ H ≤ normalizer ↑P

Next tactic:
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]
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]; 

Proof state:
p : ℕ
G : Type u_1
inst✝ : Group G
H : Subgroup G
P : Sylow p G
⊢ P ∈ fixedPoints (↥H) (Sylow p G) ↔ ∀ ⦃x : G⦄, x ∈ H → x • P = P

Next tactic:
exact Subtype.forall
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
  

Proof state:
p : ℕ
G : Type u_1
inst✝ : Group G
P : Subgroup G
hP : IsPGroup p ↥P
Q : Sylow p G
⊢ Q ∈ fixedPoints (↥P) (Sylow p G) ↔ P ≤ ↑Q

Next tactic:
rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
/-- 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
    

Proof state:
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
⊢ ∃ g, g • P = Q

Next tactic:
classical
      cases nonempty_fintype (Sylow p G)
      have H := fun {R : Sylow p G} {S : orbit G P} =>
        calc
          S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints 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 (fixedPoints Q (orbit G P)) by
        exact Exists.elim this fun R hR => by
          rw [← Sylow.ext (H.mp hR)]
          exact 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 (fixedPoints 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))⟩
/-- 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
      

Proof state:
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
⊢ ∃ g, g • P = Q

Next tactic:
cases nonempty_fintype (Sylow p G)
/-- 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)
      

Proof state:
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
⊢ ∃ g, g • P = Q

Next tactic:
have H := fun {R : Sylow p G} {S : orbit G P} =>
        calc
          S ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints 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⟩
/-- 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 ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints 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⟩
      

Proof state:
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
⊢ ∃ g, g • P = Q

Next tactic:
suffices Set.Nonempty (fixedPoints Q (orbit G P)) by
        exact Exists.elim this fun R hR => by
          rw [← Sylow.ext (H.mp hR)]
          exact R.2
/-- 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 ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints 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 (fixedPoints Q (orbit G P)) by
        

Proof state:
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
this : Set.Nonempty (fixedPoints ↥↑Q ↑(orbit G P))
⊢ ∃ g, g • P = Q

Next tactic:
exact Exists.elim this fun R hR => by
          rw [← Sylow.ext (H.mp hR)]
          exact R.2
/-- 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 ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints 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 (fixedPoints Q (orbit G P)) by
        exact Exists.elim this fun R hR => by
          

Proof state:
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
this : Set.Nonempty (fixedPoints ↥↑Q ↑(orbit G P))
R : ↑(orbit G P)
hR : R ∈ fixedPoints ↥↑Q ↑(orbit G P)
⊢ ∃ g, g • P = Q

Next tactic:
rw [← Sylow.ext (H.mp hR)]
/-- 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 ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints 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 (fixedPoints Q (orbit G P)) by
        exact Exists.elim this fun R hR => by
          rw [← Sylow.ext (H.mp hR)]
          

Proof state:
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
this : Set.Nonempty (fixedPoints ↥↑Q ↑(orbit G P))
R : ↑(orbit G P)
hR : R ∈ fixedPoints ↥↑Q ↑(orbit G P)
⊢ ∃ g, g • P = ↑R

Next tactic:
exact R.2
/-- 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 ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints 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 (fixedPoints Q (orbit G P)) by
        exact Exists.elim this fun R hR => by
          rw [← Sylow.ext (H.mp hR)]
          exact R.2
      

Proof state:
case intro
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
⊢ Set.Nonempty (fixedPoints ↥↑Q ↑(orbit G P))

Next tactic:
apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
/-- 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 ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints 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 (fixedPoints Q (orbit G P)) by
        exact Exists.elim this fun R hR => by
          rw [← Sylow.ext (H.mp hR)]
          exact R.2
      apply Q.2.nonempty_fixed_point_of_prime_not_dvd_card
      

Proof state:
case intro.hpα
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
⊢ ¬p ∣ card ↑(orbit G P)

Next tactic:
refine' fun h => hp.out.not_dvd_one (Nat.modEq_zero_iff_dvd.mp _)
/-- 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 ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints 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 (fixedPoints Q (orbit G P)) by
        exact Exists.elim this fun R hR => by
          rw [← Sylow.ext (H.mp hR)]
          exact 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 _)
      

Proof state:
case intro.hpα
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
h : p ∣ card ↑(orbit G P)
⊢ 1 ≡ 0 [MOD p]

Next tactic:
calc
        1 = card (fixedPoints 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
/-- 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 ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints 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 (fixedPoints Q (orbit G P)) by
        exact Exists.elim this fun R hR => by
          rw [← Sylow.ext (H.mp hR)]
          exact 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 (fixedPoints 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
      

Proof state:
case intro.hpα
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
h : p ∣ card ↑(orbit G P)
⊢ 1 = card ↑(fixedPoints ↥↑P ↑(orbit G P))

Next tactic:
rw [← Set.card_singleton (⟨P, mem_orbit_self P⟩ : orbit G 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 ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints 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 (fixedPoints Q (orbit G P)) by
        exact Exists.elim this fun R hR => by
          rw [← Sylow.ext (H.mp hR)]
          exact 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 (fixedPoints 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)]
      

Proof state:
case intro.hpα
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
h : p ∣ card ↑(orbit G P)
⊢ card ↑{{ val := P, property := (_ : P ∈ orbit G P) }} = card ↑(fixedPoints ↥↑P ↑(orbit G P))

Next tactic:
refine' card_congr' (congr_arg _ (Eq.symm _))
/-- 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 ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints 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 (fixedPoints Q (orbit G P)) by
        exact Exists.elim this fun R hR => by
          rw [← Sylow.ext (H.mp hR)]
          exact 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 (fixedPoints 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 _))
      

Proof state:
case intro.hpα
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
h : p ∣ card ↑(orbit G P)
⊢ fixedPoints ↥↑P ↑(orbit G P) = {{ val := P, property := (_ : P ∈ orbit G P) }}

Next tactic:
rw [Set.eq_singleton_iff_unique_mem]
/-- 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 ∈ fixedPoints R (orbit G P) ↔ S.1 ∈ fixedPoints 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 (fixedPoints Q (orbit G P)) by
        exact Exists.elim this fun R hR => by
          rw [← Sylow.ext (H.mp hR)]
          exact 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 (fixedPoints 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]
      

Proof state:
case intro.hpα
p : ℕ
G : Type u_1
inst✝¹ : Group G
hp : Fact (Nat.Prime p)
inst✝ : Finite (Sylow p G)
P Q : Sylow p G
val✝ : Fintype (Sylow p G)
H : ∀ {R : Sylow p G} {S : ↑(orbit G P)}, S ∈ fixedPoints ↥↑R ↑(orbit G P) ↔ ↑↑S = ↑R
h : p ∣ card ↑(orbit G P)
⊢ { val := P, property := (_ : P ∈ orbit G P) } ∈ fixedPoints ↥↑P ↑(orbit G P) ∧
    ∀ x ∈ fixedPoints ↥↑P ↑(orbit G P), x = { val := P, property := (_ : P ∈ orbit G P) }

Next tactic:
exact ⟨H.mpr rfl, fun R h => Subtype.ext (Sylow.ext (H.mp h))⟩
/-- 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] := by
  

Proof state:
p : ℕ
G : Type u_1
inst✝² : Group G
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fintype (Sylow p G)
⊢ card (Sylow p G) ≡ 1 [MOD p]

Next tactic:
refine' Sylow.nonempty.elim fun P : Sylow p G => _