Let $p$ be a prime number, and let $G$ be a finite group whose order is a power of $p$. Let $F$ be a field of characteristic $p$, and $V$ a nonzero vector space over $F$ equipped with a linear action of $G$. Does there exist a nonzero subspace $W \subset V$ such that G acts trivially on W?
I am stuck at this question. Now, a vector space on a finite field is again a finite field (I think so). Now, the finite field $F$ is itself a vector space over $F$. I think the group $G$ acts trivially on $F$. Is this right? Thanks beforehand.