Let G be a reductive algebraic group, P a parabolic subgroup of G with unip
otent radical P-u, and A a closed connected subgroup of P-u which is normal
ized by P. We show that P acts on A with finitely many orbits provided A is
abelian. This generalizes a well-known finiteness result, namely the case
when A is central in P-u. We also obtain an analogous result for the adjoin
t action of P on invariant linear subspaces of the Lie algebra of P-u which
are abelian Lie algebras. Finally, we discuss a connection to some work of
Mal'cev on maximal abelian subalgebras of the Lie algebra of G.