The group of endo-permutation modules

The group D(P ) of all endo-permutation modules for a finite p-group P is a finitely generated abelian group. We prove that its torsion-free rank is equal to the number of conjugacy classes of non-cyclic subgroups of P . We also obtain partial results on its torsion subgroup. We determine next the structure of Q⊗D(−) viewed as a functor, which turns out to be a simple functor SE,Q , indexed by the elementary group E of order p 2 and the trivial Out(E)-module Q . Finally we describe a rather strange exact sequence relating Q⊗D(P ) , Q⊗B(P ) , and Q⊗R(P ) , where B(P ) is the Burnside ring and R(P ) is the Grothendieck ring of QP -modules.