Non-cancellation and Mislin genus of certain groups and H0-spaces

The non-cancellation set of a group G measures the extent to which the infinite cyclic group cannot be cancelled as a direct factor of image. If G is a finitely generated group with finite commutator subgroup, then there is a group structure on its non-cancellation set, which coincides with the Hilton–Mislin genus group when G is nilpotent. Using a notion closely related to Nielsen equivalence classes of presentations of a finite abelian group, we give an alternative description of the group structure on the non-cancellation set of groups of a certain kind, and we include some computations. Analogously, we consider non-cancellation, up to homotopy, of the circle as a direct factor of a topological space. In particular, we show how the Mislin genera of certain H0-spaces with two non-vanishing homotopy groups can be identified with the genera of certain nilpotent groups.