CALCULATING THE MISLIN GENUS FOR A CERTAIN FAMILY OF NILPOTENT GROUPS

It is known that the Mislin genus of a finitely generated nilpotent group N with finite commutator subgroup admits an abelian group structure. In this paper, we compute explicitly that structure under the following additional assumptions: The torsion subgroup TN is abelian, the epimorphism N --> N/TN splits and all automorphisms of TN commute with conjugation by elements of N. Among the groups satisfying these conditions are all nilpotent split extensions of a finite cyclic group by a finitely generated free abelian group. We further prove that the function M bar-arrow-pointing-right M x N(k-1), k greater-than-or-equal-to 2, which is in general a surjective homomorphism from the genus of N onto the genus of N(k), is an isomorphism at least in an important special case. Applications to the study of non-cancellation phenomena in group theory are given.