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

For a connected Lie group G, the homotopy set C. l 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). (C) 2010 Elsevier B.V. All rights reserved.