First cohomology group for trivial group action is naturally isomorphic to group of homomorphisms

Statement
Suppose $$G$$ is a group and $$A$$ is an abelian group. Then, the fact about::first cohomology group for the trivial group action of $$G$$ on $$A$$, i.e., the group $$H^1(G,A)$$, is naturally isomorphic to the group $$\operatorname{Hom}(G,A)$$, which is the set of homomorphisms from $$G$$ to $$A$$ equipped with pointwise addition in $$A$$.

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.

Related facts

 * First homology group for trivial group action equals tensor product with abelianization