Homomorphisms from copies of group algebra to $Z$

Question

I am studying group algebras and was wondering what can be said about the following group $\hom_{\mathbb ZG}((\mathbb ZG)^n, \mathbb Z)$ for any group $G, n \in \mathbb N$. I think it should be abelian in any case. But what about generators? Are there finitely many?

Answer 1

If $A$ is any algebra, and $V$ is any left $A$-module, then $\mathrm{hom}_A(A, V)$ is naturally isomorphic to $V$, via the map that takes a function $f: A \to V$ to $f(1)$. We also have that $\mathrm{hom}_A(V \oplus W, U) \cong \mathrm{hom}_A(V, U) \oplus \mathrm{hom}_A(W, U)$ for any left $A$-modules $V, W, U$.