The mathematician Morris W.Hirsch wrote on 1977 the paper "Flat Manifolds and the Cohomology of Groups". The Theorem C establishes the following:
Theorem C: Let $G$ a nilpotent group acting linearly on a finite dimensional vector space $M$ over the field $F$. If $H^{0}(G,M)=0$, then $H^{k}(G,M)=0$, for all $k\geq 0$
I want to know if there is a counterexample when $G$ is a solvable notnilpotent group. Remember that $G$ acts linearly on $M$ it means that there is an operation $"\cdot": G\times M\rightarrow M$ such that:
(1) $g\cdot (x_{1}+x_{2})=g\cdot x_{1}+g\cdot x_{2}$, for all $g\in G; x_{1},x_{2}\in M$.
(2) $(g_{1}g_{2})\cdot x= g_{1}\cdot (g_{2}\cdot x)$, for all $g_{1},g_{2}\in G; x\in M$.
(3) $1_{G}\cdot x=x$, for all $x\in M$
(4) $g\cdot (\lambda x)=\lambda (g\cdot x)$, for all $g\in G, \lambda\in F,x\in M$
My Aproach: I have showed with an elementary proof that this Theorem C is valid for any group $G$ and $M=F$. So, I think that maybe a counterexample can be found for the case $M= F^{2}$ or $M= F^{3}$.