Wojciech Chachólski: Property of groups

A property of groups is simply a collection C of groups closed under isomorphisms. A group can have such a property (it belongs to C ) or not. We are however interested in understanding more. We would like to be able to measure how close a given group is from satisfying a given property. We would like to estimate the failure of a group to satisfy a property. How can this be done? We are going to use the notion of a C- cover of a group G. This is a homomorphism from a group X to G such that X satisfies C and this homomorphism is terminal with this respect. The aim of the talk is to classify the collection Cov(G) of all possible covers of a group G. I will show that such a classification is possible for a finite group G.

I will also present an explicit classification in the case G is finite and simple. In this case the covers can be enumerated by subsets of primes dividing the second integral homology of
G, the so- called Shur multiplier of G.

This is a joint work with E. Damian, E. Dror Farjoun, and Y. Segev.