p-radical groups are p-solvable (Q1087635)

Let k be an algebraically closed field of characteristic \(p>0\), G a finite group with Sylow p-subgroup P, and \(k_ P\) the trivial kP-module. G is called p-radical if \(k^ G_ P\) is completely reducible. It is shown that if G is p-radical then G is p-solvable. Furthermore if G is p- radical with Sylow p-subgroup P and \(D=P\cap P^ x\), then D is a vertex of some simple kG-module if \(x\in G\) and D is a defect group of some p- block of G if \(x\in C_ G(D)\).