On a conjecture of P. Landrock (Q1093004)

Let G be a finite group, k an algebraically closed field of prime characteristic p, and J the radical of kG. Landrock conjectured that \(J^ i/J^{i+1}\) is self dual as a (right) kG-module for all i. The authors show that this is false for the Mathieu group \(G=M_{11}\) of degree 11 and \(p=11\). They give another counterexample where G is a solvble group. However they show that Landrock's conjecture is true for \(G=A_ n\), \(GL_ n(g)\), \(SL_ n(q)\), and other groups.