Sylow theorems for $\infty$-groups

Matan Prasma; Tomer M. Schlank

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arXiv · 1602.04494

Sylow theorems for $\infty$-groups

Matan Prasma·Tomer M. Schlank

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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.

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BibTeX RIS

Matan Prasma, Tomer M. Schlank. 2016-02-14. Sylow theorems for $\infty$-groups. https://arxiv.org/abs/1602.04494

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