So first of all, I realized that in my sleepiness yesterday, I left off the last part of the story of why homotopy groups are groups. More precisely, we need to show that if are any pointed spaces, then
is an abelian group under the cogroup law of (the double suspension). But this group is just
and one can check that the adjointness between respects the group structure. And we showed by the Eckmann-Hilton argument yesterday that this is abelian under either the group law of or the cogroup law of ; they’re also both the same. So in particular, the homotopy classes out of a higher suspension form an abelian group. Since the homotopy groups are defined in this way, they are abelian.