Applications of group cohomology to algebra

Question

I started learning about group cohomology (of finite groups) from two books: Babakhanian and Hilton&Stammbach. The theory is indeed natural and beautiful, but I could not find many examples to its uses in algebra.

I am looking for problems stated in more classical algebraic terms which are solved elegantly or best understood through the notion of group cohomology. What I would like to know the most is "what can we learn about a finite group $G$ by looking at its cohomology groups relative to various $G$-modules?").

The one example I did find is $H^2(G,M)$ classifying extensions of $M$ by $G$.

So, my question is:

What problems on groups/rings/fields/modules/associative algebras/Lie algebras are solved or best understood through group cohomology?

Examples in algebraic number theory are also welcome (this is slightly less interesting from my current perspective, but I do remember the lecturer mentioning this concept in a basic algnt course I've taken some time ago).

Answer 1

Here's a simple example off the top of my head. A group is said to be finitely presentable if it has a presentation with finitely many generators and relations. This, in particular, implies that $H_2(G)$ is of finite rank. (You can take nontrivial coefficient systems here too.) So you get a nice necessary condition for finite presentability.

The proof of this fact is simple. If $G$ is finitely presented, you can build a finite $2$-complex that has $G$ as its fundamental group. To get an Eilenberg-Maclane space $K(G,1)$ you add $3$-cells to kill all $\pi_2$, then you add $4$-cells to kill all $\pi_3$ etc... You end up building a $K(G,1)$ with a finite $2$-skeleton.

Answer 2

In addition to what Grumpy Parsnip said about group homology, here's another application: In the field of pro-$p$-groups we have that group cohomology is an extremely useful tool for determining the structure of a group, e.g. finding the numbers of generators and relations of a pro-$p$-group:

Then the generator rank $d(G) = \dim_{\mathbb{F}_p} H^1(G,\mathbb{F}_p)$ and the relation rank $r(G) = \dim_{\mathbb{F}_p} H^2(F, \mathbb{F}_p)$ for a pro-$p$-group $G$.

There is a famous inequality discovered by Golod and Shavarefich that links those numbers above for making a statement whether a (a priori infinite) pro-$p$-group is in fact finite. This is a very beautiful and farreaching result as you can find a lot of applications in Galois theory. (Key words are: class field tower, Hilbert class field = maximal abelian unramified extension)