Let G be a finite p-group and k be a field of characteristic p. Motivated by the generating hypothesis – a deep conjecture in stable homotopy theory, we study maps between finite dimensional kG-modules that are invisible to Tate cohomology — such maps are called ghosts. A natural question is to find the smallest integer n such that the composition of any n ghosts is stably trivial, i.e., factors through a projective. I will talk about these new integer invariants for kG with a pleasant mix of techniques from group theory, representation theory, triangulated category theory and constructions motivated from homotopy theory. This is joint work with Jan Minac and Dan Christensen.