Estimating proportions of elements in finite groups of Lie type.
The authors formalize the notion of a subset of a finite group of Lie type possessing certain closure properties and prove that certain closure properties on subsets \(Q\) of a finite group \(G\) of Lie type enable the ratio \(\frac{|Q|}{|G|}\) 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. Moreover, they 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 example of application of these new methods, the authors 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. Precisely, in Section 5, the authors show the power of their main theorem (Thm. 1.3) by rederiving and extending the results of \textit{A. C. Niemeyer} and \textit{C. E. Praeger} [Proc. Lond. Math. Soc., III. Ser. 77, No. 1, 117-169 (1998; Zbl 0893.20035)] on the proportion of primitive prime divisor elements in classical groups.
scientific article; zbMATH DE number 5758111
scientific article