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 S _{E, 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. Oblatum 6-VII-1998 & 27-V-1999 / Published online: 22 September 1999