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

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Statement

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.

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.

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