Someone asked me if *all* finite abelian groups arise as homotopy groups of spheres. I strongly doubted it, and I bet ten bucks that $\mathbb{Z}_5$ is not $\pi_k(S^n)$ for any $n,k$. But I don't know how to prove it's not.

Which finite abelian groups are known to *not* arise as homotopy groups of spheres?

Is $\mathbb{Z}_5$ the smallest one? Or maybe $\mathbb{Z}_4$?

I conjecture that for no odd prime $p \gt 3$ is $\mathbb{Z}_p \cong \pi_k(S^n)$ for some $n,k$.

odd orderappearing. (But the unstable part is really outside my wheelhouse...) $\endgroup$2more comments