Sylow Theorems for Infinity Groups

Taking the fundamental group gives an equivalence of categories between pointed spaces whose homotopy groups \(\pi_n\) for \(n \neq 1\) all vanish, and groups. From this perspective, group theory can be viewed as the truncation of a more general theory of pointed connected spaces. Then the natural question is: to what extent can we do group theory in this new homotopical setting (where we don’t require the higher homotopy groups to vanish)? In this talk, we translate the Sylow theorems for finite groups to the context of finite $\infty$-groups, and use this to get a group-theoretic classification of finite nilpotent spaces, by analogy to the classification of finite nilpotent groups. We also discuss a failure of normality for $\infty$-groups, as something of a cautionary tale. Notes are available, and the primary reference is “Sylow theorems for \(\infty\)-groups” by Prasma and Schlank.