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 $H_n(G)=H_n(K(G,1))$