First cohomology group for trivial group action is naturally isomorphic to group of homomorphisms
Suppose is a group and is an abelian group. Then, the First cohomology group (?) for the trivial group action of on , i.e., the group , is naturally isomorphic to the group , which is the set of homomorphisms from to equipped with pointwise addition in .
More specifically, the isomorphism is as follows:
- The group of 1-coboundaries for the trivial group action is trivial.
- The 1-cocycles are precisely the same as the homomorphisms.
Thus, the group of 1-cocycles is naturally identified with the group of homomorphisms, and since the group of 1-coboundaries is trivial, this gives the desired identification.