Research

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Research interests

My main area of research is in group theory. I am interested in simple groups, both finite and algebraic, with a particular focus on subgroup structure, conjugacy classes and representation theory. I am also interested in permutation groups and related combinatorics, and in the application of probabilistic and computational methods.

Books and monographs



  • Classical groups, derangements and primes
    joint with M. Giudici
    Australian Mathematical Society Lecture Series, vol. 25
    Cambridge University Press, 2016, xviii+346
    ISBN: 978-1-1076-2944-8


  • Irreducible geometric subgroups of classical algebraic groups
    joint with S. Ghandour and D.M. Testerman
    Memoirs of the American Mathematical Society 239 (2016), no. 1130, vi+88
    ISBN: 978-1-4704-1494-8


  • Irreducible almost simple subgroups of classical algebraic groups
    joint with S. Ghandour, C. Marion and D.M. Testerman
    Memoirs of the American Mathematical Society 236 (2015), no. 1114, vi+110
    ISBN: 978-1-4704-1046-9

All publications and preprints

  1. Topological generation of simple algebraic groups
    joint with S. Gerhardt and R.M. Guralnick
    Journal of the European Mathematical Society (JEMS)
    87 pages, to appear

  2. On the topological generation of exceptional groups by unipotent elements
    Transformation Groups
    30 pages, to appear

  3. On the generation of simple groups by Sylow subgroups
    joint with R.M. Guralnick
    Contemporary Mathematics (Amitsur Centennial volume)
    17 pages, to appear

  4. On the maximal overgroups of Sylow subgroups of finite groups
    joint with B. Baumeister, R.M. Guralnick and H.P. Tong-Viet
    Advances in Mathematics 444 (2024), 109632, 65 pp.

  5. Strongly base-two groups
    joint with R.M. Guralnick
    Vietnam Journal of Mathematics (Tiep 60th birthday volume) 51 (2023), 657-683

  6. On the classification of extremely primitive affine groups
    joint with M. Lee
    Israel Journal of Mathematics 255 (2023), 265-282

  7. On the commuting probability of p-elements in a finite group
    joint with R.M. Guralnick, A. Moretó and G. Navarro
    Algebra and Number Theory 17 (2023), 1209-1229

  8. On soluble subgroups of sporadic groups
    Israel Journal of Mathematics 254 (2023), 313-340

  9. Normalisers of maximal tori and a conjecture of Vdovin
    joint with A.R. Thomas
    Journal of Algebra 619 (2023), 459-504

  10. On the soluble graph of a finite group
    joint with A. Lucchini and D. Nemmi
    Journal of Combinatorial Theory, Series A 194 (2023), 105708, 39 pp.

  11. On base sizes for primitive groups of product type
    joint with H.Y. Huang
    Journal of Pure and Applied Algebra 227 (2023), 107228, 43 pp.

  12. Fixed point ratios for finite primitive groups and applications
    joint with R.M. Guralnick
    Advances in Mathematics 411 (2022), 108778, 90 pp.

  13. On the Saxl graphs of primitive groups with soluble stabilisers
    joint with H.Y. Huang
    Algebraic Combinatorics 5 (2022), 1053-1087

  14. The classification of extremely primitive groups
    joint with A.R. Thomas
    International Mathematics Research Notices IMRN (2022), no. 13, 10148-10248
    [GAP and Magma code]

  15. Finite groups, minimal bases and the intersection number
    joint with M. Garonzi and A. Lucchini
    Transactions of the London Mathematical Society 9 (2022), 20-55

  16. Permutation groups with restricted stabilizers
    joint with A. Shalev
    Journal of Algebra (special issue in memory of Jan Saxl) 607 (2022), 160-185

  17. Almost elusive permutation groups
    joint with E.V. Hall
    Journal of Algebra 594 (2022), 519-543

  18. Base sizes for primitive groups with soluble stabilisers
    Algebra and Number Theory 15 (2021), 1755-1807

  19. On the involution fixity of simple groups
    joint with E. Covato
    Proceedings of the Edinburgh Mathematical Society 64 (2021), 408-426

  20. The spread of a finite group
    joint with R.M. Guralnick and S. Harper
    Annals of Mathematics 193 (2021), 619-687

  21. A note on extremely primitive affine groups
    joint with A.R. Thomas
    Archiv der Mathematik 116 (2021), 141-152

  22. On solvable factors of almost simple groups
    joint with C.H. Li
    Advances in Mathematics 377 (2021), 107499, 36 pp.

  23. Finite groups, 2-generation and the uniform domination number
    joint with S. Harper
    Israel Journal of Mathematics 239 (2020), 271-367

  24. Topological generation of exceptional algebraic groups
    joint with S. Gerhardt and R.M. Guralnick
    Advances in Mathematics 369 (2020), 107177, 50 pp.

  25. On the minimal dimension of a finite simple group
    joint with M. Garonzi and A. Lucchini (with an appendix by T.C. Burness and R.M. Guralnick)
    Journal of Combinatorial Theory, Series A 171 (2020), 105175, 32 pp.

  26. The length and depth of compact Lie groups
    joint with M.W. Liebeck and A. Shalev
    Mathematische Zeitschrift 294 (2020), 1457-1476

  27. On the Saxl graph of a permutation group
    joint with M. Giudici
    Mathematical Proceedings of the Cambridge Philosophical Society 168 (2020), 219-248

  28. On the length and depth of finite groups
    joint with M.W. Liebeck and A. Shalev (with an appendix by D.R. Heath-Brown)
    Proceedings of the London Mathematical Society 119 (2019), 1464-1492

  29. Simple groups, generation and probabilistic methods
    Groups St Andrews 2017 in Birmingham, London Mathematical Society Lecture Note Series, 455, Cambridge University Press, pp. 200-229, 2019

  30. Irreducible subgroups of simple algebraic groups - a survey
    joint with D.M. Testerman
    Groups St Andrews 2017 in Birmingham, London Mathematical Society Lecture Note Series, 455, Cambridge University Press, pp. 230-260, 2019

  31. $A_1$-type subgroups containing regular unipotent elements
    joint with D.M. Testerman
    Forum of Mathematics, Sigma 7 (2019), e12, 61 pp.

  32. On the uniform domination number of a finite simple group
    joint with S. Harper
    Transactions of the American Mathematical Society 372 (2019), 545-583
    [GAP and Magma code]

  33. The length and depth of algebraic groups
    joint with M.W. Liebeck and A. Shalev
    Mathematische Zeitschrift 291 (2019), 741-760

  34. On base sizes for almost simple primitive groups
    Journal of Algebra 516 (2018), 38-74

  35. On the involution fixity of exceptional groups of Lie type
    joint with A.R. Thomas
    International Journal of Algebra and Computation 28 (2018), 411-466

  36. Locally elusive classical groups
    joint with M. Giudici
    Israel Journal of Mathematics 225 (2018), 343-402

  37. Simple groups, fixed point ratios and applications
    Local Representation Theory and Simple Groups, EMS Series of Lectures in Mathematics, European Mathematical Society, Zürich, pp. 267-322, 2018

  38. The depth of a finite simple group
    joint with M.W. Liebeck and A. Shalev
    Proceedings of the American Mathematical Society 146 (2018), 2343-2358

  39. Generation of second maximal subgroups and the existence of special primes
    joint with M.W. Liebeck and A. Shalev
    Forum of Mathematics, Sigma 5 (2017), e25, 41 pp.

  40. On base sizes for algebraic groups
    joint with R.M. Guralnick and J. Saxl
    Journal of the European Mathematical Society (JEMS) 19 (2017), 2269-2341

  41. Permutation groups and derangements of odd prime order
    joint with M. Giudici
    Journal of Combinatorial Theory, Series A 151 (2017), 102-130

  42. On irreducible subgroups of simple algebraic groups
    joint with C. Marion and D.M. Testerman
    Mathematische Annalen 367 (2017), 1259-1309

  43. Primitive permutation groups and derangements of prime power order
    joint with H.P. Tong-Viet
    Manuscripta Mathematica 150 (2016), 255-291

  44. Classical groups, derangements and primes
    joint with M. Giudici
    Australian Mathematical Society Lecture Series, vol. 25
    Cambridge University Press, 2016, xviii+346

  45. Irreducible geometric subgroups of classical algebraic groups
    joint with S. Ghandour and D.M. Testerman
    Memoirs of the American Mathematical Society 239 (2016), no. 1130, vi+88

  46. On Pyber's base size conjecture
    joint with Á. Seress
    Transactions of the American Mathematical Society 367 (2015), 5633-5651

  47. Derangements in primitive permutation groups, with an application to character theory
    joint with H.P. Tong-Viet
    Quarterly Journal of Mathematics 66 (2015), 63-96

  48. On the prime graph of simple groups
    joint with E. Covato
    Bulletin of the Australian Mathematical Society 91 (2015), 227-240

  49. Large subgroups of simple groups
    joint with S.H. Alavi
    Journal of Algebra (special issue in memory of Ákos Seress) 421 (2015), 187-233

  50. Irreducible almost simple subgroups of classical algebraic groups
    joint with S. Ghandour, C. Marion and D.M. Testerman
    Memoirs of the American Mathematical Society 236 (2015), no. 1114, vi+110

  51. Base sizes for $\mathcal{S}$-actions of finite classical groups
    joint with R.M. Guralnick and J. Saxl
    Israel Journal of Mathematics 199 (2014), 711-756

  52. Generation and random generation: from simple groups to maximal subgroups
    joint with M.W. Liebeck and A. Shalev
    Advances in Mathematics 248 (2013), 59-95

  53. On the generating graph of direct powers of a simple group
    joint with E. Crestani
    Journal of Algebraic Combinatorics 38 (2013), 329-350

  54. On the uniform spread of almost simple linear groups
    joint with S. Guest
    Nagoya Mathematical Journal 209 (2013), 35-109

  55. Extremely primitive sporadic and alternating groups
    joint with C.E. Praeger and Á. Seress
    Bulletin of the London Mathematical Society 44 (2012), 1147-1154

  56. Extremely primitive classical groups
    joint with C.E. Praeger and Á. Seress
    Journal of Pure and Applied Algebra 216 (2012), 1580-1610

  57. Prime order derangements in primitive permutation groups
    joint with M. Giudici and R.A. Wilson
    Journal of Algebra 341 (2011), 158-178

  58. On base sizes for symmetric groups
    joint with R.M. Guralnick and J. Saxl
    Bulletin of the London Mathematical Society 43 (2011), 386-391

  59. Base sizes for sporadic simple groups
    joint with E.A. O'Brien and R.A. Wilson
    Israel Journal of Mathematics 177 (2010), 307-334

  60. On the number of prime order subgroups of finite groups
    joint with S.D. Scott
    Journal of the Australian Mathematical Society 87 (2009), 329-357

  61. Base sizes for simple groups and a conjecture of Cameron
    joint with M.W. Liebeck and A. Shalev
    Proceedings of the London Mathematical Society 97 (2009), 116-162

  62. On base sizes for actions of finite classical groups
    Journal of the London Mathematical Society 75 (2007), 545-562

  63. Fixed point ratios in actions of finite classical groups, IV
    Journal of Algebra 314 (2007), 749-788

  64. Fixed point ratios in actions of finite classical groups, III
    Journal of Algebra 314 (2007), 693-748

  65. Fixed point ratios in actions of finite classical groups, II
    Journal of Algebra 309 (2007), 80-138

  66. Fixed point ratios in actions of finite classical groups, I
    Journal of Algebra 309 (2007), 69-79

  67. Fixed point spaces in actions of classical algebraic groups
    Journal of Group Theory 7 (2004), 311-346

  68. Fixed point spaces in primitive actions of simple algebraic groups
    Journal of Algebra 265 (2003), 744-771

Theses

  1. Fixed point spaces in actions of finite and algebraic simple groups
    Ph.D. thesis, Imperial College London (2005)

  2. Reductive algebraic groups and Tits systems
    M.Sc. thesis, University of Warwick (2001)