  This is the third in a series of papers in which we prove a conjecture of
Boston and Shalev that the proportion of derangements (fixed point free
elements) is bounded away from zero for transitive actions of finite simple
groups on a set of size greater than one. This paper treats the case of
primitive subspace actions. It is also shown that if the dimension and
codimension of the subspace go to infinity, then the proportion of derangements
goes to one. Similar results are proved for elements in finite classical groups
in cosets of the simple group. The results in this paper have applications to
probabilistic generation of finite simple groups and maps between varieties
over finite fields.
