  We determine upper bounds for the maximum order of an element of a finite
almost simple group with socle T in terms of the minimum index m(T) of a
maximal subgroup of T: for T not an alternating group we prove that, with
finitely many exceptions, the maximum element order is at most m(T). Moreover,
apart from an explicit list of groups, the bound can be reduced to m(T)/4.
These results are applied to determine all primitive permutation groups on a
set of size n that contain permutations of order greater than or equal to n/4.
