Sylow theorems for ∞-groups

Matan Prasma*, Tomer M. Schlank

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Viewing Kan complexes as ∞-groupoids implies that pointed and connected Kan complexes are to be viewed as ∞-groups. A fundamental question is then: to what extent can one “do group theory” with these objects? In this paper we develop a notion of a finite ∞-group: an ∞-group whose homotopy groups are all finite. We prove a homotopical analogue of Sylow theorems for finite ∞-groups. This theorem has two corollaries: the first is a homotopical analogue of Burnside's fixed point lemma for p-groups and the second is a “group-theoretic” characterisation of finite nilpotent spaces.

Original languageAmerican English
Pages (from-to)121-138
Number of pages18
JournalTopology and its Applications
StatePublished - 15 May 2017

Bibliographical note

Publisher Copyright:
© 2017


  • Sylow subgroup
  • k-invariant
  • ∞-category
  • ∞-group


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