On the genus of nilpotent groups and spaces
- Peter Hilton
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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 .
- Charles Cassidy,Le genre d’un groupe nilpotent avec opérateurs, Comment. Math. Helv.53 (1978), 364–384. CrossRef
- Peter Hilton and Guido Mislin,On the genus of a nilpotent group with finite commutator subgroup, Math. Z.146 (1976), 201–211. CrossRef
- Peter Hilton, Guido Mislin and Joseph Roitberg,Localization of Nilpotent Groups and Spaces, North-Holland, Amsterdam, 1975.
- Guido Mislin,Nilpotent groups with finite commutator subgroups, Lecture Notes in Math.418, Springer-Verlag, Berlin, 1974, pp. 103–118.
- On the genus of nilpotent groups and spaces
Israel Journal of Mathematics
Volume 54, Issue 1 , pp 1-13
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- Peter Hilton (1) (2)
- Author Affiliations
- 1. State University of New York, Binghamton, New York, USA
- 2. Battelle Research and Studies Centers, Seattle, Washington, USA