ProCGroups.NormalSubgroups.MaximalIntersectionsAndSimpleRanks
This module sets up the finite-stage and inverse-limit description of the construction. It records the stage maps, projections, and comparison lemmas used to pass back to the completed object.
theorem maximal_normal_intersection_simple_quotient
{G : Type u} [Group G] (M : Subgroup G) [M.Normal]
(hproper : M ≠ ⊤)
(hmax : ∀ N : Subgroup G, N.Normal → M < N → N = ⊤) :
IsSimpleGroup (G ⧸ M)A maximal normal subgroup gives a simple quotient.
Show proof
by
have hnotTopLe : ¬ (⊤ : Subgroup G) ≤ M := by
intro hle
exact hproper (le_antisymm le_top hle)
rcases SetLike.not_le_iff_exists.mp hnotTopLe with ⟨g, _hgTop, hgM⟩
refine
{ exists_pair_ne := ⟨QuotientGroup.mk' M g, 1, ?_⟩
eq_bot_or_eq_top_of_normal := ?_ }
· intro hg
exact hgM ((QuotientGroup.eq_one_iff g).mp (by simpa using hg))
· intro H hH
let N : Subgroup G := Subgroup.comap (QuotientGroup.mk' M) H
have hMN : M ≤ N := by
dsimp [N]
exact QuotientGroup.le_comap_mk' M H
have hNnormal : N.Normal := by
dsimp [N]
infer_instance
by_cases hNM : N = M
· left
apply Subgroup.comap_injective (QuotientGroup.mk'_surjective M)
dsimp [N] at hNM
rw [hNM]
ext x
simp only [MonoidHom.comap_bot, QuotientGroup.ker_mk']
· right
have hMNlt : M < N := lt_of_le_of_ne hMN (by
intro hMN'
exact hNM hMN'.symm)
have hNtop : N = ⊤ := hmax N hNnormal hMNlt
apply Subgroup.comap_injective (QuotientGroup.mk'_surjective M)
dsimp [N] at hNtop
rw [hNtop, Subgroup.comap_top]Proof. Work with open normal subgroups and finite quotients of the profinite or pro-\(C\) group. Finite quotients separate points, and continuous homomorphisms are determined by their compatible quotient-level maps. Closed-subgroup and subgroup-permanence claims use ambient open-normal approximation: an open normal subgroup of the closed subgroup is refined by the intersection with an ambient open normal subgroup of \(G\). For pro-\(C\) claims, the resulting finite quotients lie in \(C\) by the closure property used in the construction: closure under quotients, subgroups, finite products, or extensions. Compatibility under refinement then assembles the finite-quotient data and proves the required profinite or pro-\(C\) statement. The quotient-level assertions are stable under passing to smaller open normal subgroups, so they define compatible data in the inverse system of finite quotients. For pro-\(C\) permanence, the construction uses only operations for which the chosen finite group class is closed. Hence the finite verification assembles to the desired profinite map, subgroup, or quotient statement. For categorical constructions such as products, pullbacks, and inverse limits, the universal property is checked by composing with the coordinate projections. The uniqueness part follows because any two candidate maps with the same finite quotient composites have the same value on every separating finite quotient. Existence is obtained by assembling the compatible coordinate maps supplied by the finite quotient construction. For quotient and subgroup claims, representatives are chosen in the ambient group and the induced map is checked to send the class of an element to the class of its image. Normality, openness, and membership in the finite quotient class are preserved by the subgroup, quotient, intersection, or inverse-image closure property being invoked.
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