In general, [A, B] does not have a group structure for arbitrary spaces A and B. However, it can be shown that [\Sigma Z,X] and [Z, \Omega X] do have natural group structures when Z and X are pointed, and the aforementioned isomorphism is of those groups. (See chapter 8, section 2) Thus, setting Z = S^{k-1} (the k-1 sphere) gives the relationship
