import
- Mathlib.GroupTheory.PGroup
- Mathlib.GroupTheory.Solvable
- ProCGroups.FreeConstructions.Framework
theorem finite_normal_subgroup_pro_p_has_central_element
(p : ℕ) [Fact p.Prime] (N : Subgroup G) :
IsFiniteSubgroup N → Nontrivial N → IsPGroup p N →
∃ z : N, z ≠ 1 ∧ z ∈ Subgroup.center NShow proof
by
intro hfinite hnontrivial hpN
letI : Finite N := hfinite
letI : Nontrivial N := hnontrivial
have hcenter : Nontrivial (Subgroup.center N) := IsPGroup.center_nontrivial hpN
obtain ⟨z, hz⟩ := exists_ne (1 : Subgroup.center N)
exact ⟨z.1, fun hz1 => hz (Subtype.ext hz1), z.2⟩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.
□theorem solvable_amalgam_criterion
(C : ProCGroups.FiniteGroupClass.{u}) :
AmalgamatedFreeProCProductData C G A B H →
IsSolvable G → IsSolvable A ∧ IsSolvable BIf an amalgamated free pro-\(C\) product embeds both factors into a solvable group \(G\), then the factors are solvable.
Show proof
by
intro hprod hG
rcases hprod with
⟨_, _, _, _, left, right, inl, inr, _hleft, _hright, hinl, hinr, _hcompat, _huniv⟩
letI : IsSolvable G := hG
exact
⟨solvable_of_solvable_injective (f := inl.toMonoidHom) hinl,
solvable_of_solvable_injective (f := inr.toMonoidHom) hinr⟩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. 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 pro-\(C\) claims, the finite quotient obtained from the construction remains in \(C\) by the relevant closure axiom, such as closure under subgroups, quotients, finite products, or extensions. The universal or lifting property is then checked on the prescribed generators, and uniqueness follows because the generators are dense or topologically generating. Consequently the two expressions have the same determining coordinates, and the defining extensionality principle for the inverse-limit, quotient, or presentation construction gives the claim in the statement.
□