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 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 excep- tions, 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. We note again that this result gives upper bounds for meo(Aut(T)) in terms of m(T), and for meo(G) in terms of m(G) (since m(T) ≤ m(G)). Moreover equality in the up- per bound meo(Aut(T)) ≤ m(T) holds when T = PSLd(q) for all but two pairs (d,q), see Table 3 and Theorem 2.16. (Theorem 2.16 and Table 3 provide good estimates for