TITLE: Radical of group algebra
QUESTION [6 upvotes]: Let $G$ be a finite group of size $p^a\cdot r$. 
Does there exist a simple way to calculate radical of the group algebra $F_p[G]$?

REPLY [2 votes]: I think the answer is "no, in general", though the terms in the question are not really well-defined. For example, if you knew the radical of the group algebra, then you could easily compute the dimension of the absolutely irreducible modules in characteristic $p.$ When $G$ is the symmetric group, this is already a notoriously difficult (and as yet unsolved) problem.