On the maximum orders of elements of finite almost simple groups and primitive permutation groups

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