  We construct natural selfmaps of compact cohomgeneity one manifolds with
finite Weyl group and compute their degrees and Lefschetz numbers. On manifolds
with simple cohomology rings this yields in certain cases relations between the
order of the Weyl group and the Euler characteristic of a principal orbit. We
apply our construction to the compact Lie group SU(3) where we extend identity
and transposition to an infinite family of selfmaps of every odd degree. The
compositions of these selfmaps with the power maps realize all possible degrees
of selfmaps of SU(3).
