  In the paper we will introduce show that the $p^3$ power map on $SU(p+t-1)$
is an H-map for $2\leq t\leq p-1$. To do this we will consider a fibration
whose base space is $SU(p+t-1)$ with the property that there is a section into
the total space. We will then use decomposition methods to identify the fibre
and the map from it to the total space. This information will be used to deduce
information about $SU(p+t-1)$. In doing this we draw together recent work of
Kishimoto and Theriault with more classical work of Cohen and Neisendorfer, and
make use of the classical theorems of Hilton and Milnor, James and Barrett.

Algebraic Topology