On the genus of nilpotent groups and spaces

Abstract

The notion ofgenus, applied to finitely generated nilpotent groups or to nilpotent spaces of finite type, was introduced by Mislin; he and the author showed how to introduce the structure of a finite abelian group into the genus if the groupN has finite commutator subgroup. An example is given of a complete genusN ₀,N ₁,...,N _s−1, which constitute a cyclic group generated byN ₁, with the additional property that eachN _i embeds in its successor as a normal subgroup with quotient cyclic of orderl; of course,N _s−1 embeds inN ₀. The construction leads to the description of a family of nilpotent spacesX ₀,X ₁,...,X _s−1, all in the same genus, no two of the same homotopy type, such that eachX _i covers its successor as a cyclicl-sheeted regular covering; of course,X _s−1 coversX ₀. Herep is a prime,n ≧ 1, ands=p ⁿ⁻¹(p−1)/2, whilel is semiprimitive modulep ⁿ.