Alternating bihomomorphism of finitely generated abelian groups arises as skew of 2-cocycle

In terms of 2-cocycles
Suppose $$G$$ and $$A$$ are finitely generated abelian groups. Suppose $$\lambda:G \times G \to A$$ is an alternating bihomomorphism of groups from $$G$$ to $$A$$. Then, there exists a 2-cocycle for trivial group action $$c:G \times G \to A$$ such that $$\operatorname{Skew}c = \lambda$$, i.e.,:

$$\! c(g,h) - c(h,g) = \lambda(g,h) \ \forall \ g,h \in G$$

In terms of cohomology groups
Suppose $$G$$ and $$A$$ are finitely generated abelian groups. Consider the homomorphism:

$$H^2(G,A) \stackrel{\operatorname{Skew}}{\to} \bigwedge^2(G,A)$$

which sends a cohomology class to the skew of any 2-cocycle representing it (that this is a homomorphism arises from the fact that skew of 2-cocycle for trivial group action of abelian group is alternating bihomomorphism). Then, this homomorphism is surjective, i.e., every alternating bihomomorphism arises from some cohomology class.

Facts used

 * 1) uses::Structure theorem for finitely generated abelian groups
 * 2) Symplectic decomposition of an alternating bilinear form taking values in a local principal ideal ring
 * 3) uses::Orthogonal direct sum of cocycles is cocycle
 * 4) Symplectic decomposition of an alternating bilinear form taking values in integers

First part:reduction to the case where $$A$$ is either infinite cyclic or cyclic of prime power order
We first show that the problem can be reduced to the case that $$A$$ is a cyclic group.

Given: Finitely generated abelian groups $$A$$ and $$G$$, an alternating bihomomorphism $$\lambda:G \times G \to A$$.

To prove: Assuming that we can solve the problem if $$A$$ were replaced by an infinite cyclic group or a cyclic group of prime power order, we can solve the problem in the general case, i.e., we can find a 2-cocycle $$c$$ such that $$\operatorname{Skew}(c) = \lambda$$.

Proof: