Commutativity of localized self-homotopy groups of symplectic groups

The self-homotopy group of a topological group G is the set of homotopy classes of self-maps of G equipped with the group structure inherited from G. We determine the set of primes p such that the p-localization of the self-homotopy group of Sp(n) is commutative. As a consequence, we see that this group detects the homotopy commutativity of p-localized Sp(n) by its commutativity almost all cases.