Let $A$ be a commutative noetherian domain of characteristic $p>0$ and $K$ be its quotient field. Let $G$ be any finite group whose order is divisible by $p$. Is $KG$ a finite-dimensional $K$-algebra?
If this is so, then I could infer that $AG$ is a classical $A$-order in $KG$, because $AG$ is finitely generated as an $A$-module (since $A$ is a noetherian ring) and we have $AG\cdot K=KG$.
I thank anybody helping.
By construction, $KG$ has $K$ dimension $|G|$, so yes, it is finite dimensional.
The group algebra $KG$ is defined by forming the free $K$-vector space using the elements of $G$ as a basis, and then enforcing the "obvious" multiplication rules to make it into an algebra. Thus if $G$ is finite, $KG$ is finite dimensional.