Exercise 16, § 6

Let G be a p-group (I, p. 72, def. 9). Put $d(G) = \dim_{\mathbf{F}_p} H^1(G, \mathbf{F}_p)$ and $r(G) = \dim_{\mathbf{F}_p} H^2(G, \mathbf{F}_p)$ (cf. exercise 15). a) Show that $d(G)$ is the minimal number of generators of G (proving that $H^1(G, \mathbf{F}_p)$ is isomorphic to the dual of $G/G^p D(G)$, and use I, p. 140, exercise 32). b) If Card (G) = $p^n$, show that one has $d(G) \leq n$ and $r(G) \leq \frac{1}{2} n(n + 1)$ (prove by induction on $n$ that there exists a presentation of G with $n$ generators and $\frac{1}{2} n(n + 1)$ relations). If $G = (\mathbf{Z}/p\mathbf{Z})^n$, show that one has $d(G) = n$ and $r(G) = \frac{1}{2} n(n + 1)$. c) If G is nontrivial, prove the inequality $r(G) > \frac{1}{4}(d(G))^2$ (« Golod-Chafarevich theorem »: use exercise 15, p. 182, and exercise 22 of VIII, § 8). d) Prove that one has $r(G) - d(G) = \dim_{\mathbf{F}_p} (H^3(G, \mathbf{Z}) \otimes_{\mathbf{Z}} \mathbf{F}_p)$.