Abstract

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 cinjugation by elements of N. Among the groups satisfying these conditions are all nilpotent split extensions of a finite cyclic group by a finitely free abelian group. We further prove that the function M ↦ M × Nk­1 k ≥ 2, which is in general a surjective homomorphism from the genus of N onto the genus of Nk , is an isomorphism at least in an imporatnt special case. Applications to the study of non-cancellation phenomena in group theory are given.

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