Prospects for Higher Group Theory

The characterisation of loop spaces as A-infinity spaces in the early 60's, suggested they constitute a homotopy-coherent analogue of groups. On the other hand, Higher Category Theory, allows us to identify loop spaces with pointed, connected 00-groupoids, or 00-groups. It is tempting to ask how much of ordinary group theory can be carried out to the homotopy-coherent framework, perhaps to constitute 'Higher Group Theory'?  

A decade ago, I tried to develop such a framework and obtained several partial results. In this talk, I'll present some of these results, like normal maps and their properties, isomorphism and Sylow theorems as well as prospects for future enquiries.