On a conjecture of Ōshima

The set of homotopy classes of self maps of a compact, connected Lie group G is a group by the pointwise multiplication which we denote by H ( G ) , and it is known to be nilpotent. Ōshima [H. Ōshima, Self homotopy group of the exceptional Lie group G 2 , J. Math. Kyoto Univ. 40 (1) (2000) 177–184] conjectured: if G is simple, then H ( G ) is nilpotent of class ⩾ rank G . We show this is true for PU ( p ) which is the first high rank example.