I am trying to find $\operatorname{Hom}_{\rm gp}((S^1)^k , (S^1)^n)$ , which is the set of continuous group homomorphisms from the $k$ dimensional torus to the $n$ dimensional torus where $1 \leqslant k \leqslant n$.
This is the hint I was given - first show that when $k = n = 1$, $\operatorname{Hom}_{\rm gp}(S^1,S^1) \cong \mathbb{Z}$
Here is what I have managed to do -
Now, for a fixed $m \in \mathbb{Z}$ one can define a map $f : S^1 \rightarrow S^1$ by $ f(z) = z^m$, but I don't know how to show that any homomorphism from $S^1$ to $S^1$ is of this form.
Assuming $\operatorname{Hom}_{\rm gp}(S^1,S^1) \cong \mathbb{Z}$ I have shown that $\operatorname{Hom}_{\rm gp}(S^1,(S^1)^n) \cong \mathbb{Z}^n$, that is, I have shown that any homomorphism from $S^1$ to $(S^1)^n$ is of the form $z \mapsto (z^{h_1}, \dots ,z^{h_n})$ for some $n$ tuple $(h_1, \dots, h_n)$ and viceversa.
My guess is that $\operatorname{Hom}_{\rm gp}((S^1)^k,(S^1)^n) \cong \mathbb{Z}^{\left( \begin{array}{c} n \\ k \end{array} \right)}$, but assuming this guess is correct I have no idea how to proceed.
Please help.