Zusammenfassung
Viewing Kan complexes as infinity-groupoids implies that pointed and connected Kan complexes are to be viewed as infinity-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 infinity-group: an infinity-group whose homotopy groups are all finite. We prove a homotopical analogue of Sylow theorems for ...
Zusammenfassung
Viewing Kan complexes as infinity-groupoids implies that pointed and connected Kan complexes are to be viewed as infinity-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 infinity-group: an infinity-group whose homotopy groups are all finite. We prove a homotopical analogue of Sylow theorems for finite infinity-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. (C) 2017 Elsevier B.V. All rights reserved.
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