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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 | import Mathlib
namespace OAI
section
noncomputable section
open Set Function
open scoped Pointwise
namespace CriticalPercolation.CompactQuotient
variable {A : Type*} [Group A]
lemma nilpotent_image_of_nilpotent_subquotient (H N : Subgroup A) [H.Normal]
[Group.IsNilpotent (N ⧸ H.subgroupOf N)] :
Group.IsNilpotent (N.map (QuotientGroup.mk' H)) := by
let f : N →* N.map (QuotientGroup.mk' H) := (QuotientGroup.mk' H).subgroupMap N
have hker : H.subgroupOf N≤f.ker := by
intro n hn
apply Subtype.ext
exact (QuotientGroup.eq_one_iff _).mpr hn
let ψ := QuotientGroup.lift (H.subgroupOf N) f hker
have hs : Surjective ψ := by
intro y
obtain ⟨n,rfl⟩ := (QuotientGroup.mk' H).subgroupMap_surjective N y
exact ⟨QuotientGroup.mk' (H.subgroupOf N) n,rfl⟩
exact Group.nilpotent_of_surjective ψ hs
lemma index_image_of_subquotient (H N : Subgroup A) [H.Normal] (hHN : H≤N) :
(N.map (QuotientGroup.mk' H)).index=N.index := by
apply N.index_map_eq QuotientGroup.mk_surjective
rw [QuotientGroup.ker_mk']
exact hHN
theorem finite_by_nilpotent_of_BGT_output (S : Set A) (hS : S.Finite) (n : ℕ)
(H N : Subgroup A) [H.Normal] (hHN : H≤N) (hH : (H : Set A)⊆S^(4*n))
[Group.IsNilpotent (N ⧸ H.subgroupOf N)] (hN : N.index≠0) :
(H : Set A).Finite ∧
∃ M : Subgroup (A ⧸ H), Group.IsNilpotent M ∧ M.index≠0 := by
have hpow : ∀ k : ℕ, (S^k).Finite := by
intro k
induction k with
| zero => simpa only [pow_zero] using (Set.finite_one : (1 : Set A).Finite)
| succ k ih => simpa only [pow_succ] using ih.mul hS
refine ⟨(hpow (4*n)).subset hH,N.map (QuotientGroup.mk' H),
nilpotent_image_of_nilpotent_subquotient H N,?_⟩
rw [index_image_of_subquotient H N hHN]
exact hN
end CriticalPercolation.CompactQuotient
end
end
end OAI
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