Israel Journal of Mathematics

, Volume 54, Issue 1, pp 1–13

On the genus of nilpotent groups and spaces

  • Peter Hilton

DOI: 10.1007/BF02764871

Cite this article as:
Hilton, P. Israel J. Math. (1986) 54: 1. doi:10.1007/BF02764871


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 genusN0,N1,...,Ns−1, which constitute a cyclic group generated byN1, with the additional property that eachNi embeds in its successor as a normal subgroup with quotient cyclic of orderl; of course,Ns−1 embeds inN0. The construction leads to the description of a family of nilpotent spacesX0,X1,...,Xs−1, all in the same genus, no two of the same homotopy type, such that eachXi covers its successor as a cyclicl-sheeted regular covering; of course,Xs−1 coversX0. Herep is a prime,n ≧ 1, ands=pn−1(p−1)/2, whilel is semiprimitive modulepn.

Copyright information

© Hebrew University 1986

Authors and Affiliations

  • Peter Hilton
    • 1
    • 2
  1. 1.State University of New YorkBinghamtonUSA
  2. 2.Battelle Research and Studies CentersSeattleUSA