Finding involutions in finite Lie type groups of odd characteristic

Abstract

Let G be a finite group of Lie type in odd characteristic defined over a field with q elements. We prove that there is an absolute (and explicit) constant c such that, if G is a classical matrix group of dimension n 2, then at least c/ log(n) of its elements are such that some power is an involution with fixed point subspace of dimension in the interval [n/3, 2n/3). If G is exceptional, or G is classical of small dimension, then, for each conjugacy class C of involutions, we find a very good lower bound for the proportion of elements of G for which some power lies in C.

Algorithms for matrix groups

“…The first step of the algorithm is to search for an element of SL(d, q) of even order that powers to a strong involution. Lübeck, Niemeyer & Praeger [80] prove the following.…”

Effective black-box constructive recognition of classical groups

“…(1 ) By [29], a strong involution i ∈ G is found after O(log d) repetitions of Step (1); thus, we expect to return to this step O(log d) times at a cost of O(log d(ξ +O +Π)). (2 ) A sample of O(d) random elements yields a Bray generator.…”

Nilpotent-independent sets and estimation in matrix algebras

“…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]. In this paper, we extend the quokka theory in a certain sense to the full matrix algebra M = M(d, q).…”