Powered by OpenAIRE graph
Found an issue? Give us feedback
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Israel Journal of Ma...arrow_drop_down
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
Israel Journal of Mathematics
Article . 1991 . Peer-reviewed
License: Springer TDM
Data sources: Crossref
image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
zbMATH Open
Article . 1991
Data sources: zbMATH Open
versions View all 2 versions
addClaim

The higher lower central series

Authors: Rosset, Shmuel;

The higher lower central series

Abstract

With every group \(G\) the author associates a family of groups \(\Gamma_ n(G)\) each of which is equal to the \(n\)-th term \(\gamma_ n(U)\) where \(U\) is a projective object in the category of epimorphisms \(U\twoheadrightarrow G\) with the kernel in the \(n\)-th centre of \(U\). It is noticed that these groups coincide with R. Baer's invariants and thus can be applied to the study of the integral homology of \(G\), as well as to some other questions in group theory. Canonically, \(\Gamma_ n(G)\twoheadrightarrow \gamma_ n(G)\) is a central extension and also, for \(m\geq n\), we have canonical homomorphisms \(\Gamma_ m(G)\twoheadrightarrow\Gamma_ n(G)\) which endow each \(\Gamma_ n(G)\) with a central filtration and enable to apply Lie ring techniques developed in this case to a certain extent. As the main application the author presents an exact bound for the rank of \(H_ 2(G)\), \(G\) finitely generated nilpotent, in terms of E. Witt's numbers for the dimensions of homogeneous components in free Lie rings. The author remarks however that this estimate is essentially due to N. Blackburn -- L. Evans, 1979. Further, there are some estimates for the order of \(\Gamma_ n(G)\) which is shown to be finite as soon as \(G\) is finite.

Related Organizations
Keywords

Homological methods in group theory, central filtration, Derived series, central series, and generalizations for groups, Associated Lie structures for groups, Baer's invariants, finitely generated nilpotent, central extension, category of epimorphisms, projective object, Lie ring, free Lie rings, integral homology, homogeneous components

  • BIP!
    Impact byBIP!
    selected citations
    These citations are derived from selected sources.
    This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    3
    popularity
    This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
    Average
    influence
    This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    Average
    impulse
    This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
    Average
Powered by OpenAIRE graph
Found an issue? Give us feedback
selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
3
Average
Average
Average
Upload OA version
Are you the author of this publication? Upload your Open Access version to Zenodo!
It’s fast and easy, just two clicks!