The odd primary H-structure of SU(n) in low rank

SU(n) has a classifying space so it is a loop space, making it homotopy associative. James and Thomas showed that a homotopy commutative Lie group must be a torus, so SU(n) is not homotopy commutative. This commutativity statement is integral. We show that after localizing at an odd prime, SU(n) is in fact homotopy commutative if n is less than or equal to (p - 1)(p - 2). We then describe some consequences.