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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | import Mathlib
import OAI.Probability.CriticalPercolation.Nilpotent.CentralCommutatorGeneration
namespace OAI
section
section
noncomputable section
open Set Function
open scoped Classical commutatorElement
namespace CriticalPercolation
variable {G A : Type*} [Group G] [Group A]
def centralCommHom (hc : commutator G≤Subgroup.center G) (a : G) :
G →* commutator G where
toFun b := ⟨⁅a,b⁆,Subgroup.commutator_mem_commutator (by trivial) (by trivial)⟩
map_one' := by apply Subtype.ext; simp
map_mul' b d := by
apply Subtype.ext
change ⁅a,b*d⁆=⁅a,b⁆*⁅a,d⁆
rw [commutatorElement_mul_right_eq_mul_conj]
have hd := hc (Subgroup.commutator_mem_commutator (H₁ := (⊤ : Subgroup G))
(H₂ := ⊤) (by trivial) (by trivial) : ⁅a,d⁆∈commutator G)
calc
⁅a,b⁆*b*⁅a,d⁆*b⁻¹ = ⁅a,b⁆*(b*⁅a,d⁆)*b⁻¹ := by group
_ = ⁅a,b⁆*(⁅a,d⁆*b)*b⁻¹ := by rw [Subgroup.mem_center_iff.mp hd b]
_ = _ := by group
theorem class_two_commutator_finite_of_cyclic_mod_torsion [Group.FG G]
(hc : commutator G≤Subgroup.center G) (ρ : G →* A)
(hker : ρ.ker≤Subgroup.center G) (g : G)
(hcyc : ∀ a : G, ∃ k : ℤ, IsOfFinOrder (ρ (a*g^(-k)))) :
(commutator G : Set G).Finite := by
let C := commutator G
let _ : CommGroup C := {
(inferInstance : Group C) with
mul_comm := fun a b => Subtype.ext (Subgroup.mem_center_iff.mp (hc b.property) a.val) }
let _ : Group.FG C := class_two_commutator_fg hc
let t := centralCommHom hc
have ht (a b : G) (hb : IsOfFinOrder (ρ b)) : IsOfFinOrder (t a b) := by
obtain ⟨m,hm,he⟩ := hb.exists_pow_eq_one
have hpow : b^m∈ρ.ker := by change ρ (b^m)=1; rwa [map_pow]
have hz := hker hpow
apply isOfFinOrder_iff_pow_eq_one.mpr
refine ⟨m,hm,?_⟩
rw [←map_pow]
apply Subtype.ext
exact commutatorElement_eq_one_iff_mul_comm.mpr (Subgroup.mem_center_iff.mp hz a)
have hself : t g g=1 := by apply Subtype.ext; simp [t,centralCommHom,commutatorElement_def]
have hga (a : G) : IsOfFinOrder (t g a) := by
obtain ⟨k,hk⟩ := hcyc a
have h := ht g (a*g^(-k)) hk
simpa only [map_mul,map_zpow,hself,one_zpow,mul_one] using h
have hag (a : G) : IsOfFinOrder (t a g) := by
have he : t a g=(t g a)⁻¹ := by
apply Subtype.ext
exact (commutatorElement_inv g a).symm
rw [he]
exact (hga a).inv
have hab (a b : G) : IsOfFinOrder (t a b) := by
obtain ⟨k,hk⟩ := hcyc b
have hw := ht a (b*g^(-k)) hk
have hg : (t a g)^k∈CommGroup.torsion C :=
(CommGroup.torsion C).zpow_mem (hag a) k
have hh := (CommGroup.torsion C).mul_mem hw hg
have he : t a (b*g^(-k))*(t a g)^k=t a b := by
rw [map_mul,map_zpow,zpow_neg]
group
rw [he] at hh
exact hh
have hsub : C≤(CommGroup.torsion C).map C.subtype := by
apply Subgroup.commutator_le.mpr
intro a _ b _
exact ⟨t a b,hab a b,rfl⟩
let _ : Finite C := CommGroup.finite_of_fg_isMulTorsion C (fun c => by
obtain ⟨d,hd,he⟩ := hsub c.property
change IsOfFinOrder d at hd
exact (show d=c from Subtype.ext he) ▸ hd)
exact Set.toFinite _
end CriticalPercolation
end
end
end
end OAI
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