The Golod–Shafarevich inequality for finite p-groups

A nontrivial finite p-group G with generator rank d(G) and relation rank r(G) = dim_{𝔽_p} H²(G, 𝔽_p) satisfies

d(G)² < 4 r(G).

So a finite p-group needs many relations: more than a quarter of the square of the number of generators.

The proof transfers a minimal presentation of G to the group algebra A = 𝔽_p[G] and applies the Golod–Shafarevich inequality for finite-dimensional algebras (TauCeti.card_sq_lt_four_mul_card_of_relations_le_span). Let π : F ↠ G be a minimal presentation by the free pro-p group F on d = d(G) generators xᵢ, and let ρ₁, …, ρ_r generate its relation subgroup R as a closed normal subgroup, with r = r(G). Write gᵢ = π xᵢ.

• The gᵢ generate G, so the elements gᵢ - 1 generate the augmentation ideal I of A as a left ideal (TauCeti.MonoidAlgebra.range_linearCombination_eq_ker_augmentation).
• Let D : F → A^d be the continuous 1-cocycle with D xᵢ = eᵢ, for the action of F on A^d by left multiplication through π (TauCeti.freeProP.exists_mem_Z1_forall_apply_of_eq). On R it is additive and conjugation-equivariant, so D R lies in the A-span S of the D ρⱼ. Modulo S, D therefore descends to a function δ on G, and extending δ linearly to A inverts the map a ↦ ∑ aᵢ (gᵢ - 1) modulo S. Hence the relation module of the gᵢ (TauCeti.MonoidAlgebra.relationModule) is contained in the span of the r vectors D ρⱼ.
• The relators lie in the Frattini subgroup of F, and composing D with the augmentation A → 𝔽_p gives a continuous homomorphism to an elementary abelian p-group, so the entries of every D ρⱼ lie in I.

Main results

• IsPGroup.sq_topologicalGeneratorRankNat_lt_four_mul_finrank_cohomFp_two: the Golod–Shafarevich inequality d(G)² < 4 r(G) for a nontrivial finite p-group G.

References

• E. S. Golod and I. R. Shafarevich, On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964).
• P. Roquette, On class field towers, in J. W. S. Cassels and A. Fröhlich (eds.), Algebraic Number Theory, Chapter IX, §4.
• J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, 2nd ed., (3.9.7).
• L. Ribes and P. Zalesskii, Profinite Groups, Theorem 7.8.5.

theorem IsPGroup.sq_topologicalGeneratorRankNat_lt_four_mul_finrank_cohomFp_two {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [Finite G] [Nontrivial G] [TopologicalSpace G] [DiscreteTopology G] (hG : IsPGroup p G) : TauCeti.topologicalGeneratorRankNat G ⋯ ^ 2 < 4 * Module.finrank (ZMod p) ↑(TauCeti.cohomFp p G 2).toModuleCat

The Golod–Shafarevich inequality. A nontrivial finite p-group G, with the discrete topology, satisfies d(G)² < 4 r(G), where d(G) is its generator rank and r(G) = dim_{𝔽_p} H²(G, 𝔽_p) is its relation rank. Finite generation follows from TauCeti.isTopologicallyFinitelyGenerated_of_fg.