Let $KG$ be a group algebra of a finite group $G$ such that the characteristic of $K$ divides the group order.
Question: When is a block of a group algebra (or even the whole group aglebra) a Koszul algebra? Is there a classification?
For example in characteristic two, the group algebra $KG$ is isomorphic to $K[x]/(x^2)$ when $G$ has two elements and here the group algebra is a Koszul algebra with Koszul dual the polynomial ring $K[x]$.
For a finite group $G$ and a field $K$ of characteristic $p$ dividing $|G|$, the group algebra $KG$ is Koszul if and only if $p=2$ and $G/O(G)$ is an elementary abelian $2$-group. This is because if $KG$ is Koszul then Ext over the cohomology ring is finite total dimension, so $H^*(G,K)$ is regular, and hence a polynomial ring. This happens if and only if $G/O(G)$ is an elementary abelian $2$-group, by a theorem I proved with Jon Carlson (MR1142778). Here, $O(G)$ denotes the largest odd order normal subgroup of $G$.