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

Suppose $G$ is a group and $A$ is an abelian group. Then, the 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 $A$.