Wedge cancellation and Mislin genus of certain suspensions

We obtain some results on non-cancellation of spheres as wedge summands of certain co-H0-spaces and Mislin genera of such spaces. The spaces concerned are cones on certain maps X→Y, such that one of the spaces is a sphere, while the other is a wedge of spheres. The main technique is the utilization of certain results on presentations of finite Z-modules, and we exploit (and compute) the group structure on the genera.