Reducing H¹ = 0 and #H² ∣ #G to subquotients of prime order

Let A be a representation of a finite group G. Suppose that for every prime p, every p-subgroup H of G and every normal subgroup N of prime index in H,

H¹(H ⧸ N, A^N) = 0    and    #H²(H ⧸ N, A^N) ∣ [H : N].

Then H¹(G, A) = 0 (isZero_groupCohomology_one_of_prime_index) and the order of H²(G, A) divides #G (natCard_groupCohomology_two_dvd_natCard_of_prime_index). No solvability of G is assumed.

This is the Sylow and tower argument by which, for the idele classes C_L of a finite Galois extension L/K of number fields, the vanishing of H¹(Gal(L/K), C_L) and the second fundamental inequality in its cohomological form #H²(Gal(L/K), C_L) ∣ [L : K] are reduced to cyclic extensions of prime degree. For those, the two hypotheses follow from the two fundamental inequalities through the Herbrand quotient.

Main statements

• TauCeti.groupCohomology.isZero_groupCohomology_one_of_prime_index: H¹(G, A) = 0.
• TauCeti.groupCohomology.natCard_groupCohomology_two_dvd_natCard_of_prime_index: the order of H²(G, A) divides #G; in particular H²(G, A) is finite.

References

• J. S. Milne, Class Field Theory, v4.03, Chapter VII, §5 (the proof of Theorem 5.1), and Chapter II, §1 (restriction to Sylow subgroups and the inflation-restriction sequence).

theorem TauCeti.groupCohomology.isZero_groupCohomology_one_of_prime_index {k G : Type u} [CommRing k] [Group G] [Finite G] (A : Rep.{u, u, u} k G) (h1 : ∀ (p : ℕ) [Fact (Nat.Prime p)] (H : Type u) [inst : Group H], IsPGroup p H → ∀ (f : H →* G), Function.Injective ⇑f → ∀ (N : Subgroup H) [inst_1 : N.Normal], Nat.Prime N.index → CategoryTheory.Limits.IsZero (groupCohomology ((Rep.res f A).quotientToInvariants N) 1)) : CategoryTheory.Limits.IsZero (groupCohomology A 1)

H¹ vanishes if it does on the subquotients of prime order. Let A be a representation of a finite group G. Suppose that for every prime p, every p-subgroup H of G (given as an injective homomorphism f : H →* G) and every normal subgroup N of prime index in H, H¹(H ⧸ N, A^N) = 0. Then H¹(G, A) = 0.

theorem TauCeti.groupCohomology.natCard_groupCohomology_two_dvd_natCard_of_prime_index {k G : Type u} [CommRing k] [Group G] [Finite G] (A : Rep.{u, u, u} k G) (h1 : ∀ (p : ℕ) [Fact (Nat.Prime p)] (H : Type u) [inst : Group H], IsPGroup p H → ∀ (f : H →* G), Function.Injective ⇑f → ∀ (N : Subgroup H) [inst_1 : N.Normal], Nat.Prime N.index → CategoryTheory.Limits.IsZero (groupCohomology ((Rep.res f A).quotientToInvariants N) 1)) (h2 : ∀ (p : ℕ) [Fact (Nat.Prime p)] (H : Type u) [inst : Group H], IsPGroup p H → ∀ (f : H →* G), Function.Injective ⇑f → ∀ (N : Subgroup H) [inst_1 : N.Normal], Nat.Prime N.index → Nat.card ↑(groupCohomology ((Rep.res f A).quotientToInvariants N) 2) ∣ N.index) : Nat.card ↑(groupCohomology A 2) ∣ Nat.card G

The order of H² divides that of the group if it does on the subquotients of prime order. Let A be a representation of a finite group G. Suppose that for every prime p, every p-subgroup H of G (given as an injective homomorphism f : H →* G) and every normal subgroup N of prime index in H, H¹(H ⧸ N, A^N) = 0 and the order of H²(H ⧸ N, A^N) divides [H : N]. Then the order of H²(G, A) divides #G; in particular H²(G, A) is finite.