2 problems

Let $V$ be a finite-dimensional vector space over a field $k$, let $G$ be a finite subgroup of $\operatorname{GL}(V)$, and let $U$ be a $G$-submodule of $V$. Write $\beta(V)$ for t…

Let $G$ be a finite group, let $V$ be a permutation $\mathbb{Z}G$-module, meaning that $V$ is a free $\mathbb{Z}$-module with a basis preserved by the action of $G$, and let…