Locally elusive classical groups

Let G be a transitive permutation group of degree n with point stabiliser H and let r be a prime divisor of n . We say that G is r -elusive if it does not contain a derangement of order r . The problem of determining the r -elusive primitive groups can be reduced to the almost simple case, and the purpose of this paper is to complete the study of r -elusivity for almost simple classical groups. Building on our earlier work for geometric actions of classical groups, in this paper we handle the remaining non-geometric actions where H is almost simple and irreducible. This requires a completely different approach, using tools from the representation theory of quasisimple groups.