Estimating proportions of elements in finite groups of Lie type

We show that certain closure properties on subsets Q of a finite group G of Lie type enable the ratio |Q|/|G| to be determined by finding the proportions of elements of Q in the maximal tori of G and the proportions of certain related subsets of the Weyl group. We prove fundamental results about these subsets Q, including those necessary for moving between these groups and their automorphism groups, normal subgroups and central quotients. As a sample application of these new techniques, we derive upper and lower bounds for the proportion of elements with order divisible by certain natural sets of primes, for the finite classical groups, thereby improving existing results.