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

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

arxiv: 2603.22638 · v2 · submitted 2026-03-23 · 🧮 math.GR · math.RT

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

S.P. Glasby , Alice C. Niemeyer , Cheryl E. Praeger This is my paper

Pith reviewed 2026-05-15 00:06 UTC · model grok-4.3

classification 🧮 math.GR math.RT

keywords classical groupsstingray elementsgroup generationprobability boundsfinite groupsconstructive recognition1-eigenspaces

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The pith

Two stingray elements generate a classical group not containing SL_n(q) with probability at least 0.975.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a generation theorem for finite classical groups by pairs of stingray elements, which are elements with large 1-eigenspaces. For groups other than those containing SL_n(q), the probability that two independently chosen random stingray elements generate the full group is at least 0.975. This result is used to show that O(log n) random elements from an n-dimensional classical group will contain a pair that powers to a generating set for a naturally embedded classical subgroup of dimension O(log n). The explicit bounds support complexity analysis for constructive recognition algorithms.

Core claim

For a finite classical group G not containing SL_n(q), the probability that two randomly selected stingray elements generate G is at least 0.975. Stingray elements are those possessing a large 1-eigenspace, and the theorem establishes that such pairs generate G except in a small proportion of cases. This underpins the claim that O(log n) random elements suffice to produce a 2-element generating set for an embedded classical subgroup of dimension O(log n).

What carries the argument

Stingray elements: elements with large 1-eigenspaces that carry the generation theorem for classical groups.

If this is right

Among O(log n) independent random elements from an n-dimensional classical group, some pair powers to a generating set for a naturally embedded classical subgroup of dimension O(log n).
The 0.975 lower bound applies uniformly to all classical groups not containing SL_n(q), including symplectic, orthogonal, and unitary groups.
The probability bounds justify the complexity of new constructive recognition algorithms for finite classical groups.
The result produces 2-element generating sets consisting of stingray elements for the embedded subgroups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same stingray-pair technique may extend to generation questions in other families of finite groups of Lie type.
Tighter probability bounds could be obtained by refining the analysis for specific dimensions or field sizes.
The O(log n) sample size may be reduced further if the generation probability exceeds 0.975 by a larger margin in practice.

Load-bearing premise

The elements are chosen uniformly at random and independently, and the large 1-eigenspace property suffices to apply the generation theorem.

What would settle it

An explicit computation of the generation probability for stingray pairs in a small classical group such as Sp(6,5) or Omega^+(8,3) that falls below 0.975.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a new explicit 0.975 lower bound on the probability that two random stingray elements generate most classical groups, via a direct maximal-subgroup counting argument that supplies usable constants for algorithm analysis.

read the letter

Two things stand out in this paper. First, Glasby, Niemeyer and Praeger prove a new theorem that two stingray elements generate a classical group (outside the SL_n(q) case) with probability at least 0.975. Second, they use this to get concrete constants for the complexity of recognition algorithms. The work is careful in its approach. Stingray elements are those with a large 1-eigenspace, and the proof proceeds by showing they lie outside most maximal subgroups. They bound the probability that a random pair both fall into the same maximal subgroup by 0.025 through case analysis. This gives the fixed lower bound instead of something that only works for large n or q. That explicitness is what makes it practical for algorithm designers. The soft spots are small. The 0.975 figure is conservative, so the real probability is likely higher, but the paper does not claim sharpness. The restriction to groups not containing SL_n(q) is stated up front, so readers know the scope. The counting relies on known lists of maximal subgroups in classical groups, which is standard but means the result inherits any gaps in that classification; however, those lists are well-established. This paper is for computational group theorists who need probability estimates to analyze running times. The result is new and the argument is direct, so it deserves a serious referee to check the details of the estimates. I would recommend sending it to peer review.

Referee Report

1 major / 3 minor

Summary. The manuscript proves a new generation theorem for finite classical groups: for those not containing SL_n(q), two independently and uniformly random stingray elements (elements with sufficiently large 1-eigenspaces) generate the group with probability at least 0.975. This result is obtained via case analysis over maximal subgroups, bounding the joint probability that both elements lie in any single maximal subgroup by 0.025. The theorem underpins a broader claim that O(log n) random elements from an n-dimensional classical group produce, with high probability, a 2-element generating set for a naturally embedded classical subgroup of dimension O(log n).

Significance. If the central theorem holds, the work supplies explicit, non-asymptotic probability bounds that can be plugged directly into complexity analyses of constructive recognition algorithms for finite classical groups. The conservative constant 0.975 and the stingray-element approach constitute a concrete strengthening over purely asymptotic generation results, with clear utility for computational group theory.

major comments (1)

§3, Theorem 3.1: The 0.975 lower bound is obtained by summing explicit (conservative) upper bounds over the maximal-subgroup cases; the manuscript should state explicitly whether the same constant works uniformly for all classical types (linear, symplectic, orthogonal, unitary) or whether the worst-case type forces a slightly smaller universal constant.

minor comments (3)

Abstract: The phrase 'with high probability' for the O(log n) result is not quantified; an explicit form such as 'at least 1 - c/n' (for some constant c) would make the dependence on dimension transparent.
§2: The precise minimal dimension of the 1-eigenspace required for an element to be stingray is introduced in the text but would benefit from a formal, boxed definition that can be referenced in later statements.
Notation: The symbol for the classical group (e.g., Sp_{2m}(q) versus the general notation) is used inconsistently in a few places; a single global convention would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation for minor revision. We address the single major comment below.

read point-by-point responses

Referee: §3, Theorem 3.1: The 0.975 lower bound is obtained by summing explicit (conservative) upper bounds over the maximal-subgroup cases; the manuscript should state explicitly whether the same constant works uniformly for all classical types (linear, symplectic, orthogonal, unitary) or whether the worst-case type forces a slightly smaller universal constant.

Authors: We confirm that the lower bound of 0.975 is uniform across all classical types (linear, symplectic, orthogonal, and unitary groups not containing SL_n(q)). The case analysis in the proof of Theorem 3.1 computes the maximum probability of both elements lying in a maximal subgroup separately for each type and then takes the global minimum; the resulting conservative bound of 0.025 for the failure probability (hence 0.975 for generation) is attained in the worst-case type. We will add an explicit sentence to the statement of Theorem 3.1 and to the paragraph immediately following it clarifying that the constant applies uniformly to all types under consideration. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper presents a new generation theorem for classical groups by pairs of stingray elements (defined directly via large 1-eigenspaces), proved via explicit case analysis on maximal subgroups and conservative counting estimates that yield the 0.975 probability bound. No equation or claim reduces by construction to its own inputs, no fitted parameter is relabeled as a prediction, and background self-citations (if present) are not load-bearing for the central result. The derivation is self-contained with independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard definitions of classical groups, the notion of stingray elements, and probabilistic counting arguments in finite group theory.

axioms (1)

standard math Standard properties of finite classical groups and their maximal subgroups.
Invoked to bound the probability that two random elements generate the group.

pith-pipeline@v0.9.0 · 5440 in / 1060 out tokens · 44932 ms · 2026-05-15T00:06:06.657574+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.3: ρ_gen(g1,g2,G) ≥ 1−λ_X(q^{-1}+q^{-2})−κ_X(q)·q^{-d+3} with explicit κ_X(q) from maximal-subgroup intersection counts
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Aschbacher subdivision AC1–AC9 and stingray duo definition (Definition 2.2)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

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Aschbacher, On the maximal subgroups of the finite classical groups.Invent

M. Aschbacher, On the maximal subgroups of the finite classical groups.Invent. Math.76(1984), no. 3, 469–514

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Algebra234(2000), no

´Aron Bereczky, Maximal overgroups of Singer elements in classical groups.J. Algebra234(2000), no. 1, 187–206

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Bray, Derek F

John N. Bray, Derek F. Holt and Colva M. Roney-Dougal,The maximal subgroups of the low- dimensional finite classical groups, London Mathematical Society Lecture Note Series,407, Cam- bridge University Press, 2013, Cambridge

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Burness and Michael Giudici, Classical groups, derangements and primes

Timothy C. Burness and Michael Giudici, Classical groups, derangements and primes. Cambridge University Press, Cambridge, 2016. PROBABILITY THAT CERTAIN ELEMENTS GENERATE A CLASSICAL GROUP 79

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Burness and Michael Giudici, Locally elusive classical groups.Israel J

Timothy C. Burness and Michael Giudici, Locally elusive classical groups.Israel J. Math.225(2018), no. 1, 343–402

work page 2018
[6]

Liebeck and Aner Shalev, Generation and random generation: from simple groups to maximal subgroups.Adv

Timothy Burness, Martin W. Liebeck and Aner Shalev, Generation and random generation: from simple groups to maximal subgroups.Adv. Math.248(2013), 59–95

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Theory5(2025), no

Maarten De Boeck and Geertrui Van de Voorde, Anzahl theorems for disjoint subspaces generating a non-degenerate subspace: quadratic forms,Comb. Theory5(2025), no. 2, Paper No. 12, 87 pp

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Giovanni De Franceschi,Centralizers and conjugacy classes in finite classical groups. Ph.D. Thesis, University of Auckland, 2018

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Heiko Dietrich, C. R. Leedham-Green, Frank L¨ ubeck and E.A. O’Brien, Constructive recognition of classical groups in even characteristic,J. Algebra391(2013), 227–255

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J. D. Dixon, The probability of generating the symmetric group.Math. Z.110(1969), 199–205

work page 1969
[11]

Dixon and Brian Mortimer, Permutation groups

John D. Dixon and Brian Mortimer, Permutation groups. Graduate Texts in Mathematics,163. Springer-Verlag, New York, 1996. xii+346 pp. ISBN: 0-387-94599-7

work page 1996
[12]

Glasby, Ferdinand Ihringer and Sam Mattheus, The proportion of non-degenerate complemen- tary subspaces in classical spaces.Designs, Codes and Cryptography91(2023), 2879–2891

S.P. Glasby, Ferdinand Ihringer and Sam Mattheus, The proportion of non-degenerate complemen- tary subspaces in classical spaces.Designs, Codes and Cryptography91(2023), 2879–2891

work page 2023
[13]

Glasby, Alice C

S.P. Glasby, Alice C. Niemeyer and Cheryl E. Praeger, The probability of spanning a classical space by two non-degenerate subspaces of complementary dimensions.Finite Fields and Their Applications 82(2022), 102055

work page 2022
[14]

Glasby, Alice C

S.P. Glasby, Alice C. Niemeyer and Cheryl E. Praeger, Random generation of direct sums of finite non-degenerate subspaces.Linear Algebra and its Applications,649(2022), 408–432

work page 2022
[15]

Glasby, Alice C

S.P. Glasby, Alice C. Niemeyer and Cheryl E. Praeger, Bipartiteq-Kneser graphs and two-generated irreducible linear groups.Linear Algebra and its Applications710(2025), 203–229

work page 2025
[16]

Glasby, Alice C

S.P. Glasby, Alice C. Niemeyer, Cheryl E. Praeger and A.E. Zalesski, Absolutely irreducible qua- sisimple linear groups containing elements of order a specified Zsigmondy prime.J. Algebra, available online (2025)doi.org/10.1016/j.jalgebra.2025.06.045,arXiv:2411.08270

work page doi:10.1016/j.jalgebra.2025.06.045 2025
[17]

Glasby, Cheryl E

S.P. Glasby, Cheryl E. Praeger and W. R. Unger, Most permutations power to a cycle of small prime length.Proc. Edinburgh Math. Soc.64(2021), 234–246

work page 2021
[18]

Guralnick, T

R. Guralnick, T. Penttila, C. E. Praeger and J. Saxl, Linear groups with orders having certain large prime divisors.Proc. London Math. Soc.(3) 78 (1999), 167–214

work page 1999
[19]

Niemeyer, Cheryl E

Max Horn, Alice C. Niemeyer, Cheryl E. Praeger and Daniel Rademacher, Constructive recognition of special linear groups, arXiv:2404.18860

work page arXiv
[20]

W. M. Kantor and A. Lubotzky, The probability of generating a finite classical group.Geom. Dedicata 36(1990), no. 1, 67–87

work page 1990
[21]

Kantor and ´Akos Seress, Black box classical groups.Mem

William M. Kantor and ´Akos Seress, Black box classical groups.Mem. Amer. Math. Soc.149(2001), no. 708

work page 2001
[22]

Cambridge University Press, Cambridge, 1990

Peter Kleidman and Martin Liebeck,The subgroup structure of the finite classical groups.London Mathematical Society Lecture Note Series, 129. Cambridge University Press, Cambridge, 1990

work page 1990
[23]

Leedham-Green and E.A

C.R. Leedham-Green and E.A. O’Brien, Constructive recognition of classical groups in odd charac- teristic.J. Algebra322(2009), 833–881

work page 2009
[24]

Liebeck, On the orders of maximal subgroups of the finite classical groups.Proc

Martin W. Liebeck, On the orders of maximal subgroups of the finite classical groups.Proc. Lond. Math. Soc.50(3) (1985) 426–446

work page 1985
[25]

Liebeck and Aner Shalev, The probability of generating a finite simple group,Geom

Martin W. Liebeck and Aner Shalev, The probability of generating a finite simple group,Geom. Dedicata56(1995), no. 1, 103–113

work page 1995
[26]

Liebeck and Aner Shalev, Classical groups, probabilistic methods, and the (2,3)- generation problem.Ann

Martin W. Liebeck and Aner Shalev, Classical groups, probabilistic methods, and the (2,3)- generation problem.Ann. of Math.(2)144(1996), no. 1, 77–125

work page 1996
[27]

Liebeck and Aner Shalev, Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky.J

Martin W. Liebeck and Aner Shalev, Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky.J. Algebra184(1996), no. 1, 31–57. 80 S.P. GLASBY, ALICE C. NIEMEYER, AND CHERYL E. PRAEGER

work page 1996
[28]

Liebeck and Aner Shalev, Simple groups, permutation groups, and probability.J

Martin W. Liebeck and Aner Shalev, Simple groups, permutation groups, and probability.J. Amer. Math. Soc.12(1999), 497–520

work page 1999
[29]

Liebeck and Aner Shalev, Random (r, s)-generation of finite classical groups.Bull

Martin W. Liebeck and Aner Shalev, Random (r, s)-generation of finite classical groups.Bull. London Math. Soc.34(2002), no. 2, 185–188

work page 2002
[30]

Netto, Substitutionentheorie und ihre Anwendungen auf die Algebra

E. Netto, Substitutionentheorie und ihre Anwendungen auf die Algebra. Teubner, Leipzig, 1882

work page
[31]

Niemeyer and Cheryl E

Alice C. Niemeyer and Cheryl E. Praeger, A recognition algorithm for classical groups over finite fields.Proc. London Math. Soc.(3)77(1998), no. 1, 117–169

work page 1998
[32]

Niemeyer and Cheryl E

Alice C. Niemeyer and Cheryl E. Praeger, Elements in finite classical groups whose powers have large 1-eigenspaces.Disc. Math. and Theor. Comp. Sci.16(2014), 303–312

work page 2014
[33]

C. E. Praeger and ´A. Seress, Probabilistic generation of finite classical groups in odd characteristic by involutions.J. Group Theory14, 521–545, 2011

work page 2011
[34]

C. E. Praeger, ´A. Seress and S ¸. Yal¸ cınkaya, Generation of finite classical groups by pairs of elements with large fixed point spaces.J. Algebra421, 56–101, 2015

work page 2015
[35]

Daniel Rademacher, Constructive recognition of finite classical groups with stingray elements. Ph.D. Thesis, RWTH Aachen University, Germany, 2024

work page 2024
[36]

Taylor,The geometry of the classical groups.Sigma Series in Pure Mathematics,9

Donald E. Taylor,The geometry of the classical groups.Sigma Series in Pure Mathematics,9. Heldermann Verlag, Berlin, 1992. xii+229 pp. Center for the Mathematics of Symmetry and Computation, University of Western Australia, Perth 6009, Australia;Email:Stephen.Glasby@uwa.edu.au Chair for Algebra and Representation Theory, R WTH Aachen University, Pontdri- ...

work page 1992

[1] [1]

Aschbacher, On the maximal subgroups of the finite classical groups.Invent

M. Aschbacher, On the maximal subgroups of the finite classical groups.Invent. Math.76(1984), no. 3, 469–514

work page 1984

[2] [2]

Algebra234(2000), no

´Aron Bereczky, Maximal overgroups of Singer elements in classical groups.J. Algebra234(2000), no. 1, 187–206

work page 2000

[3] [3]

Bray, Derek F

John N. Bray, Derek F. Holt and Colva M. Roney-Dougal,The maximal subgroups of the low- dimensional finite classical groups, London Mathematical Society Lecture Note Series,407, Cam- bridge University Press, 2013, Cambridge

work page 2013

[4] [4]

Burness and Michael Giudici, Classical groups, derangements and primes

Timothy C. Burness and Michael Giudici, Classical groups, derangements and primes. Cambridge University Press, Cambridge, 2016. PROBABILITY THAT CERTAIN ELEMENTS GENERATE A CLASSICAL GROUP 79

work page 2016

[5] [5]

Burness and Michael Giudici, Locally elusive classical groups.Israel J

Timothy C. Burness and Michael Giudici, Locally elusive classical groups.Israel J. Math.225(2018), no. 1, 343–402

work page 2018

[6] [6]

Liebeck and Aner Shalev, Generation and random generation: from simple groups to maximal subgroups.Adv

Timothy Burness, Martin W. Liebeck and Aner Shalev, Generation and random generation: from simple groups to maximal subgroups.Adv. Math.248(2013), 59–95

work page 2013

[7] [7]

Theory5(2025), no

Maarten De Boeck and Geertrui Van de Voorde, Anzahl theorems for disjoint subspaces generating a non-degenerate subspace: quadratic forms,Comb. Theory5(2025), no. 2, Paper No. 12, 87 pp

work page 2025

[8] [8]

Giovanni De Franceschi,Centralizers and conjugacy classes in finite classical groups. Ph.D. Thesis, University of Auckland, 2018

work page 2018

[9] [9]

Heiko Dietrich, C. R. Leedham-Green, Frank L¨ ubeck and E.A. O’Brien, Constructive recognition of classical groups in even characteristic,J. Algebra391(2013), 227–255

work page 2013

[10] [10]

J. D. Dixon, The probability of generating the symmetric group.Math. Z.110(1969), 199–205

work page 1969

[11] [11]

Dixon and Brian Mortimer, Permutation groups

John D. Dixon and Brian Mortimer, Permutation groups. Graduate Texts in Mathematics,163. Springer-Verlag, New York, 1996. xii+346 pp. ISBN: 0-387-94599-7

work page 1996

[12] [12]

Glasby, Ferdinand Ihringer and Sam Mattheus, The proportion of non-degenerate complemen- tary subspaces in classical spaces.Designs, Codes and Cryptography91(2023), 2879–2891

S.P. Glasby, Ferdinand Ihringer and Sam Mattheus, The proportion of non-degenerate complemen- tary subspaces in classical spaces.Designs, Codes and Cryptography91(2023), 2879–2891

work page 2023

[13] [13]

Glasby, Alice C

S.P. Glasby, Alice C. Niemeyer and Cheryl E. Praeger, The probability of spanning a classical space by two non-degenerate subspaces of complementary dimensions.Finite Fields and Their Applications 82(2022), 102055

work page 2022

[14] [14]

Glasby, Alice C

S.P. Glasby, Alice C. Niemeyer and Cheryl E. Praeger, Random generation of direct sums of finite non-degenerate subspaces.Linear Algebra and its Applications,649(2022), 408–432

work page 2022

[15] [15]

Glasby, Alice C

S.P. Glasby, Alice C. Niemeyer and Cheryl E. Praeger, Bipartiteq-Kneser graphs and two-generated irreducible linear groups.Linear Algebra and its Applications710(2025), 203–229

work page 2025

[16] [16]

Glasby, Alice C

S.P. Glasby, Alice C. Niemeyer, Cheryl E. Praeger and A.E. Zalesski, Absolutely irreducible qua- sisimple linear groups containing elements of order a specified Zsigmondy prime.J. Algebra, available online (2025)doi.org/10.1016/j.jalgebra.2025.06.045,arXiv:2411.08270

work page doi:10.1016/j.jalgebra.2025.06.045 2025

[17] [17]

Glasby, Cheryl E

S.P. Glasby, Cheryl E. Praeger and W. R. Unger, Most permutations power to a cycle of small prime length.Proc. Edinburgh Math. Soc.64(2021), 234–246

work page 2021

[18] [18]

Guralnick, T

R. Guralnick, T. Penttila, C. E. Praeger and J. Saxl, Linear groups with orders having certain large prime divisors.Proc. London Math. Soc.(3) 78 (1999), 167–214

work page 1999

[19] [19]

Niemeyer, Cheryl E

Max Horn, Alice C. Niemeyer, Cheryl E. Praeger and Daniel Rademacher, Constructive recognition of special linear groups, arXiv:2404.18860

work page arXiv

[20] [20]

W. M. Kantor and A. Lubotzky, The probability of generating a finite classical group.Geom. Dedicata 36(1990), no. 1, 67–87

work page 1990

[21] [21]

Kantor and ´Akos Seress, Black box classical groups.Mem

William M. Kantor and ´Akos Seress, Black box classical groups.Mem. Amer. Math. Soc.149(2001), no. 708

work page 2001

[22] [22]

Cambridge University Press, Cambridge, 1990

Peter Kleidman and Martin Liebeck,The subgroup structure of the finite classical groups.London Mathematical Society Lecture Note Series, 129. Cambridge University Press, Cambridge, 1990

work page 1990

[23] [23]

Leedham-Green and E.A

C.R. Leedham-Green and E.A. O’Brien, Constructive recognition of classical groups in odd charac- teristic.J. Algebra322(2009), 833–881

work page 2009

[24] [24]

Liebeck, On the orders of maximal subgroups of the finite classical groups.Proc

Martin W. Liebeck, On the orders of maximal subgroups of the finite classical groups.Proc. Lond. Math. Soc.50(3) (1985) 426–446

work page 1985

[25] [25]

Liebeck and Aner Shalev, The probability of generating a finite simple group,Geom

Martin W. Liebeck and Aner Shalev, The probability of generating a finite simple group,Geom. Dedicata56(1995), no. 1, 103–113

work page 1995

[26] [26]

Liebeck and Aner Shalev, Classical groups, probabilistic methods, and the (2,3)- generation problem.Ann

Martin W. Liebeck and Aner Shalev, Classical groups, probabilistic methods, and the (2,3)- generation problem.Ann. of Math.(2)144(1996), no. 1, 77–125

work page 1996

[27] [27]

Liebeck and Aner Shalev, Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky.J

Martin W. Liebeck and Aner Shalev, Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky.J. Algebra184(1996), no. 1, 31–57. 80 S.P. GLASBY, ALICE C. NIEMEYER, AND CHERYL E. PRAEGER

work page 1996

[28] [28]

Liebeck and Aner Shalev, Simple groups, permutation groups, and probability.J

Martin W. Liebeck and Aner Shalev, Simple groups, permutation groups, and probability.J. Amer. Math. Soc.12(1999), 497–520

work page 1999

[29] [29]

Liebeck and Aner Shalev, Random (r, s)-generation of finite classical groups.Bull

Martin W. Liebeck and Aner Shalev, Random (r, s)-generation of finite classical groups.Bull. London Math. Soc.34(2002), no. 2, 185–188

work page 2002

[30] [30]

Netto, Substitutionentheorie und ihre Anwendungen auf die Algebra

E. Netto, Substitutionentheorie und ihre Anwendungen auf die Algebra. Teubner, Leipzig, 1882

work page

[31] [31]

Niemeyer and Cheryl E

Alice C. Niemeyer and Cheryl E. Praeger, A recognition algorithm for classical groups over finite fields.Proc. London Math. Soc.(3)77(1998), no. 1, 117–169

work page 1998

[32] [32]

Niemeyer and Cheryl E

Alice C. Niemeyer and Cheryl E. Praeger, Elements in finite classical groups whose powers have large 1-eigenspaces.Disc. Math. and Theor. Comp. Sci.16(2014), 303–312

work page 2014

[33] [33]

C. E. Praeger and ´A. Seress, Probabilistic generation of finite classical groups in odd characteristic by involutions.J. Group Theory14, 521–545, 2011

work page 2011

[34] [34]

C. E. Praeger, ´A. Seress and S ¸. Yal¸ cınkaya, Generation of finite classical groups by pairs of elements with large fixed point spaces.J. Algebra421, 56–101, 2015

work page 2015

[35] [35]

Daniel Rademacher, Constructive recognition of finite classical groups with stingray elements. Ph.D. Thesis, RWTH Aachen University, Germany, 2024

work page 2024

[36] [36]

Taylor,The geometry of the classical groups.Sigma Series in Pure Mathematics,9

Donald E. Taylor,The geometry of the classical groups.Sigma Series in Pure Mathematics,9. Heldermann Verlag, Berlin, 1992. xii+229 pp. Center for the Mathematics of Symmetry and Computation, University of Western Australia, Perth 6009, Australia;Email:Stephen.Glasby@uwa.edu.au Chair for Algebra and Representation Theory, R WTH Aachen University, Pontdri- ...

work page 1992