TITLE: Essential theorems in group (co)homology
QUESTION [21 upvotes]: I'm trying to fill in the gaps in my understanding of group (co)homology and I'm wondering what are considered the "must know" theorems and concepts. I'm thinking of things along the lines of 

Hopf's formula - If $G$ has presentation $F/R$, then $H_2(G)=R \cap [F,F]/[F,R]$
If $G$ has torsion then $H_n(G)$ has no top dimension
$H_n = Tor_n$ so is the left derived functor of $\otimes$
$H^n = Ext ^n$ so is the right derived functor of $Hom$
If $G$ is discrete, then $H_n(G)=H_n(K(G,1))$

REPLY [8 votes]: Here's one which is key for calculations: Let $H$ be a subgroup of $G$ and $W_G(H) = N_G(H)/H$.  Then the restriction map $H^*(BG) \to H^*(BH)$ maps to the invariants $(H^*(BH))^{W_G(H)}$.  
When $H$ is abelian, its cohomology is well-known (polynomial tensor exterior) and thus the cohomology of $G$ is mapping to something which can in principal be computed by invariant theory. 
Follow this with Quillen's theorem that the sum of these maps over all abelian subgroups has kernel which contains only nilpotent elements, and special cases such as Milgram's theorem that this is injective for symmetric groups, and you have a powerful computational tool.
Also, here is a nice survey by Alejandro Adem (whose book with Milgram is a good reference, complementary in many ways to Brown's).  It is intended for a graduate student summer school audience:
http://www.math.uic.edu/~bshipley/ConMcohomology1.pdf

REPLY [3 votes]: It will be pretty nice to learn some Galois cohomology and Class field theory, so that you can see these machineries put to good use.