big list - Essential theorems in group (co)homology

1. 
   Interpretation of cohomology of small degree:

$H^1(G,A)$ = crossed homomorphisms $Gto A$ modulo principal ones.

$H^2(G,A)$ = equivalence classes of extensions of G by A.

$H^3(G,Center(G))$ = obstructions to existence of extensions of G by A.

2.
Transfer and its applications: If $G$ is finite then

1) $H^i(G,M)$ is a torsion group annihilated by multiplication by $|G|$.

2) Embedding of $p$-primary component of $H^i(G,M)$ into a subgroup of $H^i(P,M)$, for any $p$-Sylow subgroup $Psubset G$.

3.
In general, Brown's book "Cohomology of groups" gives a decent overview of what is good to know.

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