SCHWEITZER 2014 Problem 6

Let $\rho:G\to GL(V)$ be a representation of a finite $p$ -group $G$ over a field of characteristic $p$ . Prove that if the restriction of the linear map $\sum_{g\in G} \rho(g)$ to a finite dimensional subspace $W$ of $V$ is injective, then the subspace spanned by the subspaces $\rho(g)W$ $(g\in G)$ is the direct sum of these subspaces.