Exercise 9, § 16

Let G be a finite group and M a $\mathbf{Z}$[G]-module. Denote by $N_G$ the homomorphism $m\mapsto \sum_ggm$ from M to M; its image is contained in the submodule $M^G$ of M consisting of the invariant elements. Let $T_G(M)$ be the quotient group $M^G/N_G(M)$. a) Let $c\in Z^2(G,M)$. Prove that the element $\sum_hc(h, g)$ belongs to $M^G$; we denote its class in $T_G(M)$ by $\theta_c(g)$. Prove that $\theta_c$ is a homomorphism from G to $T_G(M)$ that is zero when $c\in B^2(G,M)$. Deduce a homomorphism $\theta_M: H^2(G,M)\rightarrow$ Hom(G$,T_G(M)$) from it.

b) Let $u: M\rightarrow N$ be a homomorphism of $\mathbf{Z}$[G]-modules and $\widetilde{u}: T_G(M)\rightarrow T_G(N)$ the homomorphism induced by $u$. Prove that we have $\theta_N\circ H^2(u) =$ Hom(1$_G,\widetilde{u}$)$\circ \theta_M$. c) Assume from now on that G is cyclic; let $\sigma$ be a generator of G, and denote its order by $n$. Let $\chi \in$ Hom(G$,\mathbf{Z}/n\mathbf{Z}$) be the homomorphism that sends $\sigma$ to the class of 1, and let $\lambda \in H^2(G,\mathbf{Z})$ be its image by the connecting homomorphism associated with the exact sequence $0\rightarrow \mathbf{Z}\rightarrow \mathbf{Z}\rightarrow \mathbf{Z}/n\mathbf{Z}\rightarrow 0$. Prove that $\lambda$ is the class of the element $\varepsilon \in Z^2(G,\mathbf{Z})$ such that for $0\leqslant p < n$ and $0\leqslant q < n$, the image $\varepsilon (\sigma^p, \sigma^q)$ is equal to 0 if $p+q < n$ and to 1 if $p+q\geqslant n$. For $m\in M^G$, denote by $h_m$ the $\mathbf{Z}$[G]-linear mapping from $\mathbf{Z}$ to M such that $h_m(1) =m$. Prove that the image of $H^2(h_m)(\lambda )$ by $\theta_M$ is equal to the class of $m$, so that $\theta_M$ is surjective.

d) Prove that $\theta_M$ is an isomorphism. (Let $c\in Z^2(G,M)$. Prove that we may assume $c(g,1) =c(1, g) = 1$ for every $g\in G$, and determine the images by $d^1$ of the elements $b$ and $a_m(m\in M)$ of $C^1(G,M)$ defined by $b(\sigma^p) =\sum^{p-1}_{i=0}c(\sigma^i, \sigma )$ and $a_m(\sigma^p) =\sum^{p-1}_{i=0}\sigma^im.$)