ProCGroups.Topologies.Conjugation
This module supplies the topological part of the construction. It checks continuity and stagewise neighborhood properties so that the completed object inherits the required topology.
noncomputable def conjNormalContinuousMulEquiv
{G : Type u} [TopologicalSpace G] [Group G] [IsTopologicalGroup G]
(N : Subgroup G) [N.Normal] (g : G) : N ≃ₜ* N :=
{ toMulEquiv := MulAut.conjNormal g
continuous_toFun := by
apply Continuous.subtype_mk
change Continuous fun x : N => g * (x : G) * g⁻¹
simpa [mul_assoc] using
((continuous_const : Continuous fun _ : N => g).mul continuous_subtype_val).mul
(continuous_const : Continuous fun _ : N => g⁻¹)
continuous_invFun := by
apply Continuous.subtype_mk
change Continuous fun x : N => g⁻¹ * (x : G) * (g⁻¹)⁻¹
simpa [mul_assoc] using
((continuous_const : Continuous fun _ : N => g⁻¹).mul continuous_subtype_val).mul
(continuous_const : Continuous fun _ : N => (g⁻¹)⁻¹) }Conjugation by an ambient element as a continuous automorphism of a normal subgroup.
@[simp] theorem conjNormalContinuousMulEquiv_apply
{G : Type u} [TopologicalSpace G] [Group G] [IsTopologicalGroup G]
(N : Subgroup G) [N.Normal] (g : G) (x : N) :
N.conjNormalContinuousMulEquiv g x = MulAut.conjNormal g xThe conjugation equivalence is evaluated by conjugating representatives.
Show proof
rflProof. 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. Continuity is checked by composing with all finite quotient projections; each composite is a continuous map between finite or profinite quotient spaces. 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. The topological assertion is checked by the initial topology of the inverse limit. After composing with each finite-stage projection, the relevant map is a continuous finite-stage homomorphism or an operation on a finite product; compactness, Hausdorffness, total disconnectedness, and profiniteness are then inherited from the finite stages by the standard inverse-limit argument. 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.
□noncomputable def quotientConjugationOnTopologicallyCharacteristicQuotient
{G : Type u} [TopologicalSpace G] [Group G] [IsTopologicalGroup G]
(N : Subgroup G) [N.Normal] (K : Subgroup N) [K.TopologicallyCharacteristic]
(hNactsTrivially :
∀ n x : N, (((MulAut.conjNormal (n : G)) x) * x⁻¹ : N) ∈ K) :
(G ⧸ N) →* MulAut (N ⧸ K) := by
let hKchar : K.TopologicallyCharacteristic := inferInstance
letI : K.Normal := by infer_instance
let prequotientAction : G →* MulAut (N ⧸ K) :=
{ toFun := fun g =>
hKchar.quotientMulEquiv (Subgroup.conjNormalContinuousMulEquiv (G := G) N g)
map_one' := by
ext a
obtain ⟨x, rfl⟩ := QuotientGroup.mk'_surjective K a
rw [Subgroup.TopologicallyCharacteristic.quotientMulEquiv_mk]
simp only [Subgroup.conjNormalContinuousMulEquiv, map_one, ContinuousMulEquiv.coe_mk, MulAut.one_apply,
QuotientGroup.mk'_apply]
map_mul' := by
intro g h
ext a
obtain ⟨x, rfl⟩ := QuotientGroup.mk'_surjective K a
change
QuotientGroup.mk' K
((Subgroup.conjNormalContinuousMulEquiv (G := G) N (g * h)) x) =
hKchar.quotientMulEquiv (Subgroup.conjNormalContinuousMulEquiv (G := G) N g)
(hKchar.quotientMulEquiv
(Subgroup.conjNormalContinuousMulEquiv (G := G) N h) (QuotientGroup.mk' K x))
rw [Subgroup.TopologicallyCharacteristic.quotientMulEquiv_mk,
Subgroup.TopologicallyCharacteristic.quotientMulEquiv_mk]
congr 1
ext
simp only [Subgroup.conjNormalContinuousMulEquiv, map_mul, ContinuousMulEquiv.coe_mk, MulAut.mul_apply,
MulAut.conjNormal_apply, mul_assoc]}
have hNker : N ≤ prequotientAction.ker := by
intro g hg
ext a
obtain ⟨x, rfl⟩ := QuotientGroup.mk'_surjective K a
change
QuotientGroup.mk' K ((MulAut.conjNormal g) x) =
QuotientGroup.mk' K x
exact
(QuotientGroup.eq_iff_div_mem (N := K)
(x := (MulAut.conjNormal g) x) (y := x)).2
(by simpa [div_eq_mul_inv] using hNactsTrivially ⟨g, hg⟩ x)
exact QuotientGroup.lift N prequotientAction hNkerTopological variant of \(\mathrm{ProCGroups.GroupTheory.quotientConjugationOnCharacteristicQuotient}\). It only needs \(K\) to be preserved by continuous automorphisms, since the conjugation maps used to define the action are continuous.
@[simp] theorem quotientConjugationOnTopologicallyCharacteristicQuotient_mk_apply_mk
{G : Type u} [TopologicalSpace G] [Group G] [IsTopologicalGroup G]
(N : Subgroup G) [N.Normal] (K : Subgroup N) [K.TopologicallyCharacteristic]
(hNactsTrivially :
∀ n x : N, (((MulAut.conjNormal (n : G)) x) * x⁻¹ : N) ∈ K)
(g : G) (n : N) :
quotientConjugationOnTopologicallyCharacteristicQuotient
(G := G) N K hNactsTrivially
(QuotientGroup.mk' N g) (QuotientGroup.mk' K n) =
QuotientGroup.mk' K ((MulAut.conjNormal g) n)The topological conjugation action sends representatives to conjugates.
Show proof
by
dsimp [quotientConjugationOnTopologicallyCharacteristicQuotient]
rflProof. 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. Continuity is checked by composing with all finite quotient projections; each composite is a continuous map between finite or profinite quotient spaces. 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. The topological assertion is checked by the initial topology of the inverse limit. After composing with each finite-stage projection, the relevant map is a continuous finite-stage homomorphism or an operation on a finite product; compactness, Hausdorffness, total disconnectedness, and profiniteness are then inherited from the finite stages by the standard inverse-limit argument.
□