Let $G$ be a finite group, and $V$ a (say complex) finite dimensional representation of $G$.
Let me view $V$ as a $G$-module in the obvious way. Is it true that $$H^n(G;V)=0$$ for $n\geq 1$? I suspect that the answer is no, but I haven't been able to find a counterexample. If the answer (surprisingly) turns out to be yes, I would love to see a proof!