Let p be an odd prime. We describe the homotopy fiber of the linearization map from Waldhausen's A(*) to Quillen's K(Z) in the range of degrees where the J-homomorphism is onto, i.e. up to degree 2(p-1)p-2. As a corollary we find p-torsion families in the stable pseudoisotopy space of a point beginning in degree 4p-4. We also get information about the fiber in the range of degrees where the S^1-transfer map to Q(S^0) is onto, i.e. up to degree 2p(p+2)(p-1)-2. Finally we discuss a possible model for the homotopy type of the fiber.