, Volume 54, Issue 1, pp 1-13

On the genus of nilpotent groups and spaces

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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 .