Sylow theorems for ∞-groups

Abstract

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.