## Exciting news!

Tim Penttila and I are currently finishing our new book on projective geometry (over the real numbers) that will be published by Cambridge University Press.

Solutions to selected exercises

## Published papers

Some of the following appear on the open preprint server arxiv.org.

1. Non-free points for groups generated by a pair of 2 x 2 matrices,
Journal of the London Mathematical Society (2) 62 (2000), 795–801.
2. A note on torsion groups generated by pairs of 2 × 2 rational parabolic matrices,
Bulletin of the Australian Mathematical Society 61 (2000), 151–152,
with Grant Cairns.
3. The crystallographic restriction, permutations, and Goldbach’s Conjecture,
American Mathematical Monthly 110 (2003), 202–209,
with Grant Cairns and Devin Kilminster.
4. Finite permutation groups with a transitive minimal normal subgroup,
Proceedings of the London Mathematical Society (3) 89 (2004), 71–103,
with Cheryl E. Praeger.
Designs, Codes and Cryptography 32 (2004), 9–14,
with Simeon Ball, Michel Lavrauw, and Tim Penttila
6. Totally Goldbach numbers and related conjectures,
Australian Mathematical Society. Gazette 31 (2004), 251–255,
with David van Golstein Brouwers and Grant Cairns.
7. Bounds and quotient actions of innately transitive groups,
Journal of the Australian Mathematical Society 79 (2005), 95–112.
8. Transitive eggs,
Innovations in Incidence Geometry 4 (2006), 1–12.
with Tim Penttila.
9. Tight sets and m-ovoids of finite polar spaces,
Journal of Combinatorial Theory Series A 114 (2007), no. 7, 1293–1314.
with Shane Kelly, Maska Law, and Tim Penttila.
10. Transitive decompositions of graph products: rank 3 product action type,
Journal of Group Theory 11 (2008), no. 2, 185–228.
with Geoffrey Pearce and Cheryl E. Praeger.
11. A construction of one-dimensional affine flag-transitive linear spaces,
Finite Fields and their Applications 14 (2008), no. 2, 537-548,
with Michael Pauley.
12. Overgroups of cyclic Sylow subgroups of linear groups,
Communications in Algebra 36 (2008), no. 7, 2503–2543,
with Tim Penttila.
13. Elation generalised quadrangles for which the number of lines on a point is the successor of a prime,
Journal of the Australian Mathematical Society 85 (2008), no. 3, 289–303,
with Tim Penttila and Csaba Schneider.
14. A hemisystem of a nonclassical generalised quadrangle,
Designs, Codes and Cryptography 51 (2009), no. 2, 157–165,
with Frank De Clerck and Nicola Durante.
15. A classification of transitive ovoids, spread, and m-systems of polar spaces,
Forum Mathematicum 21 (2009), no. 2, 181–216,
with Tim Penttila.
16. Tight sets and m-ovoids of generalised quadrangles,
Combinatorica 29 (2009), no. 1, 1–17,
with Maska Law and Tim Penttila.
17. Every flock generalised quadrangle has a hemisystem,
Bulletin of the London Mathematical Society 42 (2010), no. 5, pp. 795–810,
with Michael Giudici and Gordon Royle. arxiv
18. A geometric construction of Mathon’s perp-system from four lines of PG(5, 3),
Journal of Combinatorial Designs 18 (2010), no. 6, pp. 450–461,
with Frank De Clerck.
19. Point regular groups of automorphisms of generalised quadrangles,
Journal of Combinatorial Theory Series A, 118 (2011), no. 6, pp. 450–461,
with Michael Giudici.
20. Intriguing sets of partial quadrangles,
Journal of Combinatorial Designs 19 (2011), no. 3, pp. 217–245,
with Frank De Clerck and Nicola Durante.
21. Generalised quadrangles with a group of automorphisms acting primitive on points and lines,
Journal of Combinatorial Theory Series A 119 (2012), no. 7, pp. 1479 – 1499,
with Michael Giudici, Joy Morris, Gordon Royle and Pablo Spiga.
22. Weighted intriguing sets of finite generalised quadrangles,
Journal of Algebraic Combinatorics 36 (2012), no. 1, pp. 149 – 173,
with Alice Devillers and Jeroen Schillewaert.
23. A classification of the strongly regular generalised Johnson graphs,
Annals of Combinatorics 16 (2012), no. 3, pp. 489 – 506,
with Andrew D. Cannon and Cheryl E. Praeger.
24. Low dimensional models of the finite split Cayley hexagon,
Contemporary Mathematics 579 (2012), pp. 1 – 20,
with Nicola Durante.
25. Hemisystems of small flock quadrangles,
Designs, Codes and Cryptography 67 (2013), no. 1, pp. 137 – 157,
with Michael Giudici and Gordon Royle.
26. An alternative way to generalise the pentagon,
Journal of Combinatorial Designs 21 (2013), no. 4, pp. 163 – 179,
with Simeon Ball, Alice Devillers and Klara Stokes.
27. The classification of almost simple 3/2-transitive groups,
Transactions of the American Mathematical Society 365 (2013), no. 8, pp. 4257 – 4311,
with Michael Giudici, Martin Liebeck, Cheryl Praeger, and Jan Saxl.
28. Unitals in the Desarguesian projective plane of order 16,
Journal of Statistical Planning and Inference 144 (2014), pp. 110 – 122.
with Anton Betten, Cheryl Praeger and Alfred Wasserman.
29. Bounds on the diameter of Cayley graphs of the symmetric group,
Journal of Algebraic Combinatorics 40 (2014), no. 1, pp. 1 – 22.
with Nick Gill, Thomas P. Hayes, Harald A. Helfgott, Ákos Seress and Pablo Spiga.
30. Editorial: Special issue on finite geometries in honor of Frank De Clerck,
Designs, Codes and Cryptography 72 (2014), no. 1, pp. 1 – 5,
with Jan De Beule, Nicola Durante, and Michael Lavrauw.
31. AS-configurations and skew-translation generalised quadrangles,
Journal of Algebra (special issue in memory of Ákos Seress) 421 (2015), pp. 311 – 330.
with S. P. Glasby and Eric Swartz.
32. Triple factorisations of the general linear group and their associated geometries,
Linear Algebra and its Applications 469 (2015), pp. 169–203,
with Seyed Hassan Alavi and Cheryl E. Praeger.
33. Locally triangular graphs and rectagraphs with symmetry,
Journal of Combinatorial Theory Series A 133 (2015), pp. 1 – 28,
with Alice Devillers, Joanna B. Fawcett, and Cheryl E. Praeger.
34. Completing Segre’s proof of Wedderburn’s little theorem,
Bulletin of the London Mathematical Society 47 (2015), no. 3, pp. 483 – 492.
with Tim Penttila.
35. An enumeration of certain projective ternary two-weight codes,
Journal of Combinatorial Designs, 24 (2016), no. 1, pp. 21 – 35,
with Michael Martis and Sylvia Morris.
36. Generalised quadrangles and transitive pseudo-hyperovals,
Journal of Combinatorial Designs, 24 (2016), no. 4, pp. 151 – 164,
with S. P. Glasby, Tomasz Popiel, and Cheryl E. Praeger.
37. A note on relative hemisystems of Hermitian generalised quadrangles,
Designs, Codes & Cryptography, 81 (2016), no. 1, pp. 131 – 144,
with Melissa Lee and Eric Swartz.
38. Point-primitive generalised hexagons and octagons,
Journal of Combinatorial Theory Series A, 147 (2017), pp. 186 – 204,
with S. P. Glasby, Tomasz Popiel, Cheryl Praeger, and Csaba Schneider.
39. New non-existence proofs for ovoids of Hermitian polar spaces and hyperbolic quadrics,
Annals of Combinatorics, 21 (2017), no 1, pp. 25 – 42,
with Jan De Beule and Ferdinand Ihringer.
40. Point-primitive, line-transitive generalised quadrangles of holomorph type,
Journal of Group Theory 20 (2017), no. 2, pp. 269 – 287.
with Tomasz Popiel and Cheryl Praeger.
41. A relative m-cover of the Hermitian surface is a relative hemisystem,
Journal of Algebraic Combinatorics 45 (2017), no. 4, pp. 1217 – 1228.
with Melissa Lee.
42. Synchronization and separation in the Johnson schemes,
Portugaliae Mathematica 74 (2017), no. 3, pp. 213 – 232.
with Mohammed Aljohani and Peter J. Cameron.
43. A classification of finite antiflag-transitive generalized quadrangles,
Transactions of the American Mathematical Society 370 (2018), no. 3, pp. 1551 – 1601,
with Cai-Heng Li and Eric Swartz. doi
44. Bruck nets and partial Sherk planes,
Journal of the Australian Mathematical Society 104 (2018), no. 1, pp. 1 – 12.
with Joanna B. Fawcett and Jesse Lansdown. doi
45. A new infinite family of hemisystems of the Hermitian surface,
Combinatorica 38 (2018), no. 1, pp. 43 – 66.
with Melissa Lee, Koji Momihara, Qing Xiang. doi
46. On m-ovoids of regular near polygons,
Designs, Codes & Cryptography 86 (2018), no. 5, pp. 997 – 1006.
with Jesse Lansdown and Melissa Lee. doi
47. On regular induced subgraphs of bipartite cages,
Journal of Combinatorial Theory Series A 158 (2018), pp. 254 – 275.
with Anurag Bishnoi and Gordon Royle. doi
48. Maximal linear groups induced on the Frattini quotient of a p-group,
Journal of Pure and Applied Algebra 222 (2018), no. 10, pp. 2931 – 2951,
with S. P. Glasby, Luke Morgan, and Alice Niemeyer.
49. On the circular altitude of a graph,
Australasian Journal of Combinatorics 72 (2018), no. 2, pp. 357 – 368,
with Brian Corr, Alice Devillers, Daniel Hawtin, Irene Pivotto, and Eric Swartz. doi
50. On abstract ovals with Pascalian secant lines,
Journal of Group Theory 21 (2018), no. 6, 1051 – 1064,
with Tim Harris and Tim Penttila. doi
51. The covering radii of the 2-transitive unitary, Suzuki, and Ree groups,
Journal of Group Theory 22 (2019), no. 1, 103 – 117,
with Cheryl E. Praeger and Binzhou Xia. doi
52. Simple groups, product actions, and generalized quadrangles,
Nagoya Mathematical Journal 234 (2019), 87 – 126.
with Tomasz Popiel and Cheryl Praeger. doi
53. On intriguing sets of the Penttila-Williford association scheme,
Linear Algebra and its Applications 582 (2019), no. 1 (December), 327 – 345.
with Klaus Metsch. doi
54. Corrigendum to “Maximal linear groups induced on the Frattini quotient of a p-group’’ [J. Pure Appl. Algebra 222 (10) (2018) 2931–2951].
Journal of Pure and Applied Algebra 223 (2019), no. 12, 5428 – 5429.
with S. P. Glasby, Luke Morgan, Alice C. Niemeyer.
55. Permutations with orders coprime to a given integer,
Electronic Journal of Combinatorics, 27 (2020), #P1.6,
with Stephen Glasby, Scott Harper, Cheryl E. Praeger. doi
56. On $p$-groups with automorphism groups related to the Chevalley group $G_2(p)$,
Journal of the Australian Mathematical Society 108 (2020), no. 3, pp. 321 – 331. doi
with Saul D. Freedman and Luke Morgan.
57. The thickness of Schubert cells as incidence structures,
Journal of the Australian Mathematical Society 109 (2020), no. 2, pp. 145 – 156.
with Arun Ram and Jon Xu. doi
58. No almost simple sporadic group acts primitively on the points of a generalised quadrangle,
Discrete Mathematics 344 (2021), no. 4, 112291.
with James Evans. doi
59. A classification of finite locally 2-transitive generalized quadrangles,
Transactions of the American Mathematical Society 374 (2021), no. 3, 1535 – 1578.
with Cai-Heng Li and Eric Swartz. doi
60. Pseudo-ovals of elliptic quadrics as Delsarte designs of association schemes,
Linear Algebra and its Applications 624 (2021), pp. 281–317,
with Giusy Monzillo and Alessandro Siciliano. doi
61. Partial linear spaces with a rank 3 affine primitive group of automorphisms,
Journal of the London Mathematical Society 104 (2021), no. 3, pp. 1011–1084,
with Alice Devillers, Joanna B. Fawcett, Cheryl E. Praeger. doi
62. Correction to: New non-Existence proofs for ovoids of Hermitian polar spaces and hyperbolic quadrics.
Annals of Combinatorics 25 (2021), pp. 1095–1097,
with Jan De Beule and Ferdinand Ihringer doi
63. Orbits of Sylow subgroups of finite permutation groups,
Journal of Algebra 607 (2022), Part A, pp. 107–133,
with Alexander Bors, Alice Devillers, Michael Giudici, Cheryl Praeger, and Gordon Royle. doi
64. Synchronising primitive groups of diagonal type exist,
Bulletin of the London Mathematical Society 54 (2022), pp. 1131–1144,
with Michael Giudici, Jesse Lansdown, Gordon Royle. doi

## To appear

• The minimum degree of minimal Ramsey graphs for cliques,
Bulletin of the London Mathematical Society,
with Anurag Bishnoi and Thomas Lesgourges. doi
• Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry, Algebraic Combinatorics,
with Jesse Lansdown.

## Submitted

• Codes and designs in Johnson graphs from symplectic actions on quadratic forms,
with Alice Devillers, Mark Ioppolo, Cheryl E. Praeger.
• Simpler foundations for the hyperbolic plane,
with Tim Penttila.
• FinInG: a package for finite incidence geometry,
with Anton Betten, Philippe Cara, Jan De Beule, Max Neunhoeffer, and Michel Lavrauw.

## In preparation

• Rank 3 separating graphs, with Michael Giudici, Jesse Lansdown, and Gordon Royle.
• Euclidean geometry is incidence geometry, with Tim Penttila.
• On m-ovoids of dual twisted triality hexagons
• Triple factorisations of classical groups by maximal parabolic subgroups and their associated geometries, with Seyed Hassan Alavi and Cheryl E. Praeger.
• On the elation group of the dual of the $H(4,q^2)$ generalised quadrangle, with Michel Lavrauw and Tim Penttila.
• On the geometry of Kantor, with Nicola Durante.
• Classical geometry I. Real Euclidean and Inversive Geometry, with Tim Penttila [book]
• Classical geometry II. Real Affine Geometry, with Tim Penttila [book]
• Classical geometry IV. Foundations of Euclidean geometry. with tim Penttila [book]
• Incidence geometry I. Affine and projective geometry over a field. (with Tim Penttila [book]
• Incidence geometry II. Foundations of affine and projective geometry. (with Tim Penttila) [book]
• Incidence geometry III. Finite geometries. (with Tim Penttila) [book]