The probability that two elements with large $1$-eigenspaces generate a classical group

TL;DR

This paper proves that randomly chosen elements in finite classical groups often generate large subgroups, with high probability, using elements called stingray elements with large 1-eigenspaces.

Contribution

It introduces a new theorem on generating classical groups with stingray elements and provides explicit probability bounds for group generation.

Findings

High probability of generating classical subgroups with O(log n) elements

Explicit probability bounds for generation are at least 0.975

Results support complexity analysis of recognition algorithms

Abstract

With high probability, among $O(\log n)$ independent randomly selected elements from a finite $n$-dimensional classical group, some pair of elements power to a $2$-element generating set for a naturally embedded classical subgroup of dimension $O(\log n)$. The $2$-element generating set produced consists of certain elements with large $1$-eigenspaces, called stingray elements. Underpinning this result is a new theorem on the generation of a finite classical group by a pair of stingray elements. In particular we show that, for classical groups not containing ${\rm SL}_n(q)$, the probability of generation is at least $0.975$. The explicit probability bounds we obtain will be applied to justify complexity analyses for new constructive recognition algorithms for finite classical groups.