  Let V be a d-dimensional vector space over a field of prime order p. We
classify the affine transformations of V of order at least p^d/4, and apply
this classification to determine the finite primitive permutation groups of
affine type, and of degree n, that contain a permutation of order at least n/4.
Using this result we obtain a classification of finite primitive permutation
groups of affine type containing a permutation with at most four cycles.
