For a fixed group G I will define the group (co)homology of G with coefficients in an arbitrary G-module. After this I will calculate a few examples, and compute the low-dimensional (co)homology groups of G. These low-dimensional groups are related to: Group extensions of G, derivations of G-modules, the abelianization of G, universal central extensions of G, crossed G-modules, Schur multipliers, and more. I will say as much about these topics as I can fit into one hour.
An introduction to group (co)homology