  Viewing Kan complexes as $\infty$-groupoids implies that pointed and
connected Kan complexes are to be viewed as $\infty$-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 $\infty$-group: an $\infty$-group
with finitely many non-trivial homotopy groups which are all finite. We prove a
homotopical analog of the Sylow theorems for finite $\infty$-groups. We derive
two corollaries: the first is a homotopical analog of the Burnside's fixed
point lemma for $p$-groups and the second is a "group-theoretic"
characterization of (finite) nilpotent spaces.
