On the commutativity of the localized self homotopy groups of SU(n)

For a connected Lie group G, the homotopy set [G,G] inherits the group structure by the pointwise multiplication and is called by the self homotopy group of G. In this paper we work with the case G=SU(n),U(n). It was shown by McGibbon that SU(n) and U(n) themselves are homotopy commutative when they are localized at p and p>2n−1. Thus the p-localized self homotopy groups of SU(n) and U(n) are commutative, if p>2n−1. Then the converse is true?In this paper, we completely determine, for which p, the p-localized self homotopy group of G is commutative, in the case G=U(n),SU(n) or SU(n)/H where H is a subgroup of the center of SU(n).