The semigroup of the homotopy classes of the self-homotopy maps of a finite complex
which induce the trivial homomorphism on homotopy groups is nilpotent. We
determine the nilpotency of these semigroups of compact Lie groups and finite Hopf
spaces of rank 2. We also study the nilpotency of semigroups for Lie groups of higher
rank. Especially, we give Lie groups with the nilpotency of the semigroups arbitrarily
large.
