Homotopy of spheres $\pi_{n+1}(S^n) \simeq \mathbb{Z}_2$

Question

I have a problem: I have to prove that $$ \pi_{n+1}(S^n) \simeq \mathbb{Z}/2\mathbb{Z} $$ when $n \ge 3$. I know the Freudenthal suspension theorem and the Hopf fibration. Is there an easy method to do this?

Answer 1

Hint: Consider the Hopf map $p:S^3\to S^2$, representing the generator of $\pi_3(S^2)$. The suspended map $\Sigma p: S^4\to S^3$ represents the generator of $\pi_4(S^3)$.