Estimating proportions of elements in finite groups of Lie type

Abstract

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.

Effective black-box constructive recognition of classical groups

“…Every semisimple element in G = SX d (q) lies in a maximal torus; the structure of these tori is known, see for example [36,Sec. 3].…”

“…We now summarise easy but important consequences of properties of ppd(q, e) elements as discussed in [35]; to obtain the stated proportions, using [36], we count the number of tori (up to conjugacy) with suitable direct factors. …”

On proportions of pre-involutions in finite classical groups

“…where the sum is over all F -conjugacy classes C ⊂ W and This strategy has been used in [8], and since developed in a general setting in [9].…”

Nilpotent-independent sets and estimation in matrix algebras

“…The 'quokka theory' of Niemeyer and Praeger [23] is a method for estimating the cardinality of subsets Q of finite simple groups of Lie type such that Q is a union of conjugacy classes and membership of Q depends only on the semisimple part of the Jordan decomposition of an element. This technique was first used by Lehrer [12,13] to study representations of finite Lie-type groups and has recently proven useful for several estimation problems [14,20,21].…”