ProCGroups.GroupTheory.Conjugation
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.
import
- Mathlib.GroupTheory.QuotientGroup.Basic
noncomputable def quotientMulEquiv
(hH : H.Characteristic) (e : G ≃* G) :
G ⧸ H ≃* G ⧸ H := by
letI : H.Normal := by infer_instance
exact QuotientGroup.congr H H e (Subgroup.characteristic_iff_map_eq.mp hH e)An automorphism descends to the quotient by a characteristic subgroup.
@[simp] theorem quotientMulEquiv_mk
(hH : H.Characteristic) (e : G ≃* G) (g : G) :
hH.quotientMulEquiv e (QuotientGroup.mk' H g) = QuotientGroup.mk' H (e g)The quotient equivalence induced by a characteristic subgroup sends representatives to 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. 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.
□noncomputable def quotientConjugationOnCharacteristicQuotient
{G : Type u} [Group G]
(N : Subgroup G) [N.Normal] (K : Subgroup N) [K.Characteristic]
(hNactsTrivially :
∀ n x : N, (((MulAut.conjNormal (n : G)) x) * x⁻¹ : N) ∈ K) :
(G ⧸ N) →* MulAut (N ⧸ K) := by
let hKchar : K.Characteristic := inferInstance
letI : K.Normal := by infer_instance
let prequotientAction : G →* MulAut (N ⧸ K) :=
{ toFun := fun g => hKchar.quotientMulEquiv (MulAut.conjNormal g)
map_one' := by
ext a
obtain ⟨x, rfl⟩ := QuotientGroup.mk'_surjective K a
rw [Subgroup.Characteristic.quotientMulEquiv_mk]
simp only [map_one, 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 ((MulAut.conjNormal (g * h)) x) =
hKchar.quotientMulEquiv (MulAut.conjNormal g)
(hKchar.quotientMulEquiv (MulAut.conjNormal h) (QuotientGroup.mk' K x))
rw [Subgroup.Characteristic.quotientMulEquiv_mk,
Subgroup.Characteristic.quotientMulEquiv_mk]
congr 1
ext
simp only [map_mul, 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 hNkerIf \(K\) is characteristic in a normal subgroup \(N\) and inner conjugation by elements of \(N\) is trivial on \(N/K\), then \(G/N\) acts on \(N/K\) by conjugation.
@[simp] theorem quotientConjugationOnCharacteristicQuotient_mk_apply_mk
{G : Type u} [Group G]
(N : Subgroup G) [N.Normal] (K : Subgroup N) [K.Characteristic]
(hNactsTrivially :
∀ n x : N, (((MulAut.conjNormal (n : G)) x) * x⁻¹ : N) ∈ K)
(g : G) (n : N) :
quotientConjugationOnCharacteristicQuotient
(G := G) N K hNactsTrivially
(QuotientGroup.mk' N g) (QuotientGroup.mk' K n) =
QuotientGroup.mk' K ((MulAut.conjNormal g) n)The algebraic conjugation action sends representatives to conjugates.
Show proof
by
dsimp [quotientConjugationOnCharacteristicQuotient]
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. 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. 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.
□