Endomorphism algebras of transitive permutation modules for p-groups

Let G be a finite p-group with subgroup H and k a field of characteristic p. We study the endomorphism algebra E = End_kG(k_H ↑^G), showing that it is a split extension of a nilpotent ideal by the group algebra kN_G(H)/H. We identify the space of endomorphisms that factor through a projective kG-module and hence the endomorphism ring of k_H ↑^G in the stable module category, and determine the Loewy structure of E when G has nilpotency class 2 and G, H] is cyclic. Received: 3 November 2008