Motivation
A general paradigm in group theory is that a group G should be studied via its group representations. A slight generalization of those representations are the G-modules: a G-module is an abelian group M together with a group action of G on M, with every element of G acting as an automorphism of M. In the sequel we will write G multiplicatively and M additively.
Given such a G-module M, it is natural to consider the subgroup of G-invariant elements:
Now, if N is a submodule of M (i.e. a subgroup of M mapped to itself by the action of G), it isn't in general true that the invariants in M/N are found as the quotient of the invariants in M by those in N: being invariant 'modulo N ' is broader. The first group cohomology H1(G,N) precisely measures the difference. The group cohomology functors H* in general measure the extent to which taking invariants doesn't respect exact sequences. This is expressed by a long exact sequence.
Read more about this topic: Group Cohomology
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