The odd primary $H$-structure of low rank Lie groups and its application to exponents

A compact, connected, simple Lie group $G$ localized at an odd prime $p$ is shown to be homotopy equivalent to a product of homotopy associative, homotopy commutative spaces, provided the rank of $G$ is low. This holds for $SU(n)$, for example, if $n\leq (p-1)(p-3)$. The homotopy equivalence is usually just as spaces, not multiplicative spaces. Nevertheless, the strong multiplicative features of the factors can be used to prove useful properties, which after looping can be transferred multiplicatively to $\Omega G$. This is applied to prove useful information about the torsion in the homotopy groups of $G$, including an upper bound on its exponent.