Understanding Hatcher's corollary 4.25

Question

Here he proved the cyclic groups is infinite by considering the degree, but the degree is from $H^n(S^n)\to H^n(S^n)$, how is this related to the homotopy group $\pi_n$?. Moreover, how does $z\to z^k$ has degree $k$ imply $\pi_n(S^n)\to\mathbb{Z}$ is an isomorphism?

Answer 1

He use the Hurewicz theorem to identify $\pi_n(S^n)$ to $H_n(S^n)$ https://en.wikipedia.org/wiki/Hurewicz_theorem