Question on group homomorphisms involving the standard Z-basis

Question

Let $(e_1, ..., e_n)$ be the standard $\Bbb Z$-basis of $\Bbb Z^n$. Let $x_1,..., x_n$ be elements of an abelian group $G$. Prove that there exists a homomorphism $f$ $:$ $\Bbb Z^n$ $\rightarrow$ $G$ such that $f$$(e_i)$ $=$ $x_i$ for all $i$.

Can anyone please help me out here?

Answer 1

Hint:

Define $f:\Bbb{Z}^n\rightarrow G$ by $$a_1e_1+\dots +a_ne_n \mapsto a_1x_1 +\dots +a_nx_n$$ for $a_1e_1+\dots +a_ne_n\in \Bbb{Z}_n$

Try to verify that
(i) $f$ is a homomorphism
(ii) $f(e_i)=x_i$