Non-abelian Group Cohomology
Using the G-invariants and the 1-cochains, one can construct the zeroth and first group cohomology for a group G with coefficients in a non-abelian group. Specifically, a G-group is a (not necessarily abelian) group A together with an action by G.
The zeroth cohomology of G with coefficients in A is defined to be the subgroup
of elements of A fixed by G.
The first cohomology of G with coefficients in A is defined as 1-cocycles modulo an equivalence relation instead of by 1-coboundaries. The condition for a map φ to be a 1-cocycle is that and if there is an a in A such that . In general, is not a group when A is non-abelian. It instead has the structure of a pointed set – exactly the same situation arises in the 0th homotopy group, which for a general topological space is not a group but a pointed set. Note that a group is in particular a pointed set, with the identity element as distinguished point.
Using explicit calculations, one still obtains a truncated long exact sequence in cohomology. Specifically, let
be a short exact sequence of G-groups, then there is an exact sequence of pointed sets
Read more about this topic: Group Cohomology
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