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On families of nilpotent subgroups and associated coset posets

Abstract

We study some properties of the coset poset associated with the family of subgroups of class \(\le 2\) of a nilpotent group of class \(\le 3\). We prove that under certain assumptions on the group the coset poset is simply-connected if and only if the group is 2-Engel, and 2-connected if and only if the group is nilpotent of class 2 or less. We determine the homotopy type of the coset poset for the group of \(4\times 4\) upper unitriangular matrices over \(\mathbb {F}_p\), and for the Burnside groups of exponent 3.

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Acknowledgements

BV acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 682922), as well as support from Universidad Nacional Autónoma de México (UNAM) under the programme “Becas de Posdoc DGAPA.” This project was also supported by the Danish National Research Foundation through the Copenhagen Centre for Geometry and Topology (GEOTOP-DNRF151).

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Correspondence to Bernardo Villarreal.

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Communicated by Chuck Weibel.

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Gritschacher, S., Villarreal, B. On families of nilpotent subgroups and associated coset posets. J. Homotopy Relat. Struct. (2022). https://doi.org/10.1007/s40062-022-00315-w

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  • DOI: https://doi.org/10.1007/s40062-022-00315-w

Keywords

  • Nilpotent group
  • 2-Engel group
  • Colimit of groups
  • Coset poset
  • Higher generation
  • Simplicial set
  • Simplicial complex

Mathematics Subject Classification

  • Primary 57M07
  • 20F18
  • 55U10
  • Secondary 20F12
  • 20F45