I read the following fact in a paper: If $K$ is a finite $p$ group and $M$ a discrete $\mathbb{Z}_p[K]$ module such that $M^K=0$ then $M=0$.
This seems obviously wrong to me, which leads me to believe I am not understanding the definitions correctly. The obvious counterexample is $K=M=\mathbb{F}_p$. In this case $M$ has no $K$ invariants and yet is not zero.
Actually this is not a counterexample at all, since to be a module we must consider the finite field acting on itself via multiplication, and therefore 1 is an invariant.
In general, the question boils down to showing under the circumstances any p-adic representation of a p group contains the trivial representation. Any ideas?