Platonic and alternatinc 2-groups

Abstract

We recall Schur's work on universal central extensions and develop the analogous theory for categorical extensions of groups. We prove that the String 2-groups are universal in this sense and study in detail their restrictions to the finite subgroups of the Spin groups. Of particular interest are subgroups of the 3-sphere Spin(3), as well as the spin double covers of the alternating groups, whose categorical extensions turn out to be governed by the stable 3-stem.