  The main purpose of this paper is to prove a group-theoretic generalization
of a theorem of Katz on isocrystals. Along the way we reprove the
group-theoretic generalization of Mazur's inequality for isocrystals due to
Rapoport-Richartz, and generalize from split groups to unramified groups a
result of Kottwitz-Rapoport which determines when an affine Deligne-Lusztig
subset of the affine Grassmannian is non-empty.
