>>12462819If the group is discrete, then there's not too much to be extracted from . It pretty much only tells us that is an Eilenberg-MacLane space . However, should we be interested in group (co)homology, this would now become quite important, as one has , where the stuff on the left hand side is the group (co)homology of with coefficients in a suitable module. This isn't an easy thing to compute in general, though, so a little trickery can be used. Assume is finite. Then its cohomology splits into a direct sum of pieces, one for each prime divisor of the order of the group. Choosing the coefficients to be in the field for each such prime, we kill everything except the -component, and this gives us .
This is when we reintroduce homotopy theory. If you have ever heard of the Bousfield-Kan completion, this is it. Since was assumed finite, it is "-good" for every prime, that is, , where denotes the -completion of and all cohomology is with -coefficients. We now know that the cohomology of the completed classifying space is isomorphic to the group cohomology of our group (with the same coefficients!). Furthermore, this completion comes with a nice property: iff , so we may simplify our considerations by applying this. Suppose we have a finite group such that and a much more approachable classifying space. Then !
