Sylow theorems for $\infty$-groups

TL;DR

This paper develops a homotopical analog of Sylow theorems for finite $g$-groups, extending classical group theory concepts to the realm of $g$-groupoids and homotopy theory.

Contribution

It introduces finite $g$-groups and proves Sylow theorems in a homotopical setting, bridging group theory and higher homotopy structures.

Findings

Homotopical Sylow theorems for finite $g$-groups

Homotopical Burnside's fixed point lemma for $p$-groups

Characterization of finite nilpotent spaces

Abstract

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.