Exercise 5, § 16

Let G be a group and M a not necessarily abelian group endowed with a homomorphism $\varphi : G\rightarrow$ Aut(M); for $g\in G$ and $m\in M$, set $^gm=\varphi (g)(m)$. Denote by $H^0(G,M)$ the subgroup of M consisting of the elements fixed by the action of G; its identity element is called the marked or distinguished element of $H^0(G,M)$. Let $Z^1(G,M)$ be the set of mappings $c: G\rightarrow M$ satisfying $c(gh) =c(g)^gc(h)$ for $g, h$ in G. For $c\in Z^1(G,M)$ and $m\in M$, the mapping $g\mapsto m c(g) (^gm)^{-1}$ belongs to $Z^1(G,M)$; this defines an action of the group M on $Z^1(G,M)$. The quotient set is denoted by $H^1(G,M)$; it is endowed with a marked element, namely the class of the constant mapping with value 1. If M is abelian, then the set $H^1(G,M)$ coincides with that defined in Exercise 1.

a) Let N be a second group endowed with an action of G and $u: M\rightarrow N$ a group homomorphism compatible with the actions of G. For $i= 0,1$, define homomorphisms $H^i(u) : H^i(G,M)\rightarrow H^i(G,N)$ that send the marked element of $H^i(G,M)$ to that of $H^i(G,N)$.

b) Let $0\rightarrow M'\rightarrow^{\iota}M\rightarrow^{\pi}M''\rightarrow 0$ be an exact sequence of groups with operators in G. Let $x''\in H^0(G,M'')$, and let $x$ be an element of M such that $\pi (x) =x''$. Prove that there exists an element $c'\in Z^1(G,M')$ such that $\iota \circ c'(g) =g^{-1}c(g)$ for every $g\in G$ and that the class of $c'$ in $H^1(G,M')$ depends only on $x''$; we denote that class by $\partial (x'')$. Prove that the sequence

is exact (a sequence of mappings $G\rightarrow^iF\overset{p}{\rightarrow}E$, where the set E is endowed with a marked element $e$, is called exact in F if Im($i$) $=p^{-1}(e)$).

c) Suppose, moreover, that $M'$ is contained in the center of M. Construct a mapping $\partial^1: H^1(G,M'')\rightarrow H^2(G,M')$ such that the inverse image of the identity element by $\partial^1$ is the image of $H^1(\pi )$. Also define an action of the group $H^1(G,M')$ on the set $H^1(G,M)$ such that two elements are conjugate for this action if and only if they have the same image by $H^1(\pi )$.

d) Let K be a commutative field, L a finite Galois extension of K with Galois group G, and $n$ an integer. Let G act on the group $\mathbf{G}\mathbf{L}_n(L)$ by setting $^{\sigma}A= (\sigma (a_{ij}))$ for $\sigma \in G$ and A $= (a_{ij})\in \mathbf{G}\mathbf{L}_n(L)$. Deduce from V, §10, No. 5, p. 64, Proposition 9 that the set $H^1(G,\mathbf{G}\mathbf{L}_n(L))$ is reduced to one element (“Hilbert’s 90 theorem”).