TITLE: Use of noncommutative group cohomology
QUESTION [7 upvotes]: I have seen a lot of places where the group cohomology when a group acts on a module, is extensively used. But beyond seeing the definition and some claims of partial results, I havent seen any uses of the case when we replace action on a module by action on noncommutative group. It seems hard to believe that much can be made out of it as H_1 is just a set, not even a group. Can somebody explain some uses of bringing up and studying this notion?

REPLY [6 votes]: Matt E's response is the canonical first answer.  As someone (else) who works with nonabelian group (in particular, Galois) cohomology frequently, let me give a second answer.
Certain interesting maps between commutative cohomology groups are defined via a non-commutative intermediary.  The justification for this is that, though one does not in general have anything like a "long exact sequence" in non-commutative cohomology, if one has an extension of $\mathfrak{g}$-modules
$$1 \rightarrow Z \rightarrow G \rightarrow A \rightarrow 1$$
where $A$ is commutative and $Z$ is central, then one gets a connecting map in cohomology 
$$\Delta: H^1(\mathfrak{g},A) \rightarrow H^2(\mathfrak{g},Z).$$
Often elements in an $H^2$ may be viewed as "obstructions" to something desirable happening at the $H^1$-level.  In particular, this is the case for the period-index obstruction map in the Galois cohomology of abelian varieties.  See for instance 
http://math.uga.edu/~pete/wc1.pdf
and also publications [12], [14], [16] on
http://math.uga.edu/~pete/papers.html