Let $V$ be a vector space of dimension $n$ over a finite field $\mathbb{F}$, and let $G$ be a subgroup of the finite group $\operatorname{GL}(V)$. Then $G$ acts on the graded algebra $\mathbb{F}(V)$ consisting of polynomial functions on $V$. Then the subalgebra of invariants by $\mathbb{F}[V]^G$ is known to be a finitely generated, graded, connected algebra of Krull dimension $n$ over $\mathbb{F}$.
In this setting, if there are homogeneous invariants $f_1,\ldots,f_n$ such that $\prod_{i=1}^n\deg(f_i)=|G|$ and $\dim(\operatorname{F}[V]^G/(f_1,\ldots,f_n))=0$, then it is known that $\mathbb{F}[V]^G=\mathbb{F}[f_1,\ldots,f_n]$.
More generally, suppose that we have only the condition that $\dim(\mathbb{F}[V]^G/(f_1,\ldots,f_n))=0$. Then are the following statements true?
- $|G|$ divides $\prod_{i=1}^n\deg(f_i)$.
- $\mathbb{F}[V]^G$ is a finitely generated free module over $\mathbb{F}[f_1,\ldots,f_n]$ of rank $\prod_{i=1}^n\deg(f_i)/|G|$.
In my explorations of small matrix groups, I seem to be finding both of these statements are true. There must be some general result of this nature.
Thanks for any references or help here.