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

S. P. Glasby, Alice C. Niemeyer, Cheryl E. Praeger

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. For example, we show that, for classical groups not containing $\SL_n(q)$, the probability of generation is at least $0.983$ except for one exceptional case. The explicit probability bounds we obtain will be applied to justify complexity analyses for new constructive recognition algorithms for finite classical groups.

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

Abstract

With high probability, among independent randomly selected elements from a finite -dimensional classical group, some pair of elements power to a -element generating set for a naturally embedded classical subgroup of dimension . The -element generating set produced consists of certain elements with large -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. For example, we show that, for classical groups not containing , the probability of generation is at least except for one exceptional case. The explicit probability bounds we obtain will be applied to justify complexity analyses for new constructive recognition algorithms for finite classical groups.
Paper Structure (26 sections, 44 theorems, 178 equations, 19 tables)

This paper contains 26 sections, 44 theorems, 178 equations, 19 tables.

Table of Contents

  1. Introduction
  2. A brief commentary on Theorem \ref{['thm:main']}
  3. Naturally embedded subgroups for other finite simple groups
  4. Groups and stingray elements
  5. Properties of Stingray Elements
  6. Stingray duos in classical groups
  7. Strategy for proving Theorem \ref{['thm:main']}
  8. Bounds on numbers of subspaces and duos
  9. Estimates for the linear type L
  10. Estimates for all classical types X not L
  11. Asch1 Stabilisers of subspaces
  12. Reducible duos for type
  13. Reducible duos for classical types
  14. Asch2: Stabilisers of decompositions
  15. Asch3 Stabilisers of extension fields
  16. ...and 11 more sections

Key Result

Theorem 1.1

Suppose that $G$ is a classical group of type $({\mathbf X}, n, q)$ with $n > 8$ as in Table tab:G. Then for each $\eta > 0$, there is a positive constant $k(\eta)$ such that, with probability at least $1-\eta$, among $k(\eta) \log n$ independent uniformly distributed random elements from $G$, some

Theorems & Definitions (88)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Lemma 1.4
  • proof
  • Lemma 1.5
  • proof
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • ...and 78 more