Let $K$ be a field of $\operatorname{char}= p>0$ , let $G$ be finite group of order $p$, and $V$ is non zero $KG$-module.
How do I show that there exist non-zero $v\in V$ such that $gv=v $ for all $g\in G$, and how do I show that all the irreducible $KG$-modules are isomorphic, if that makes sense.
The first question makes sense for sure!