Tomasz Popiel

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My research publications are listed below (roughly organised by topic), together with some errata and addenda.

The links provided below point (mostly) to the journal versions of my papers. If you cannot access these, feel free to contact me.

arXiv versions also exist in most cases, but these may differ slightly from the journal versions.

Related links: my Google Scholar profile, and my PhD thesis.

Finite group theory (and related topics)

1. Conjugacy class fusion from four maximal subgroups of the Monster (with A. Pisani). arXiv preprint. Associated Python code.
2. Recognisability of the sporadic groups by the isomorphism types of their prime graphs (with M. Lee). arXiv preprint.
3. The maximal subgroups of the Monster (with H. Dietrich and M. Lee). arXiv preprint. Associated Python code.
4. Saxl graphs of primitive affine groups with sporadic point stabilisers (with M. Lee), International Journal of Algebra and Computation 33 (2023) 369-389. Article.
5. M, B and Co₁ are recognisable by their prime graphs (with M. Lee), Journal of Group Theory 26 (2023) 193-205. Article; erratum (typos only).
6. An exotic presentation of Q₂₈ (with W. H. Mannan), Algebraic & Geometric Topology 21 (2021) 2065-2084. Article.
7. On the second-largest Sylow subgroup of a finite simple group of Lie type (with S. P. Glasby and A. C. Niemeyer), Bulletin of the Australian Mathematical Society 99 (2019) 203-211. Article.
8. Finding involutions with small support (with A. C. Niemeyer), Bulletin of the Australian Mathematical Society 94 (2016) 43-47. Article.
9. Nilpotent-independent sets and estimation in matrix algebras (with B. P. Corr and C. E. Praeger), London Mathematical Society Journal of Computation and Mathematics 18 (2015) 404-418. Article.
10. Abundant p-singular elements in finite classical groups (with A. C. Niemeyer and C. E. Praeger), Journal of Algebra 408 (2014) 189-204. Article.
11. Algorithms to identity abundant p-singular elements in finite classical groups (with A. C. Niemeyer and C. E. Praeger), Bulletin of the Australian Mathematical Society 86 (2012) 50-63. Article; erratum (typos only).
12. On semiregular permutations of a finite set (with A. C. Niemeyer, C. E. Praeger and S. Yalcinkaya), Mathematics of Computation 81 (2012) 605-622. Article.
13. On proportions of pre-involutions in finite classical groups (with A. C. Niemeyer and C. E. Praeger), Journal of Algebra 324 (2010) 1016-1043. Article; addendum.
14. Straight-line programs with memory and matrix Bruhat decomposition (with A. C. Niemeyer and C. E. Praeger). arXiv preprint only. Associated GAP code by Daniel Rademacher.

Algebraic geometry over finite fields

1. Combinatorial invariants for nets of conics in PG(2, q) (with M. Lavrauw and J. Sheekey), Designs, Codes and Cryptography 90 (2022) 2021-2067. Article.
2. Solids in the space of the Veronese surface in even characteristic (with N. Alnajjarine and M. Lavrauw), Finite Fields and their Applications 83 (2022) 102068. Article.
3. Nets of conics of rank one in PG(2, q), q odd (with M. Lavrauw and J. Sheekey), Journal of Geometry 111 (2020) 36. Article.
4. The symmetric representation of lines in PG(F³ ⊗ F³) (with M. Lavrauw), Discrete Mathematics 343 (2020) 111775. Article.

Finite incidence geometry

1. Simple groups, product actions, and generalised quadrangles (with J. Bamberg and C. E. Praeger), Nagoya Mathematical Journal 234 (2019) 87-126. Article.
2. Point-primitive, line-transitive generalised quadrangles of holomorph type (with J. Bamberg and C.E.Praeger), Journal of Group Theory 20 (2017) 269-287. Article.
3. Point-primitive generalised hexagons and octagons (with J. Bamberg, S. P. Glasby, C. E. Praeger and C. Schneider), Journal of Combinatorial Theory, Series A 147 (2017) 186-204. Article.
4. Generalised polygons admitting a point-primitive almost simple group of Suzuki or Ree type (with L. Morgan), Electronic Journal of Combinatorics 23 (2016) #P1.34. Article; erratum.
5. Generalised quadrangles and transitive pseudo-hyperovals (with J. Bamberg, S. P. Glasby and C. E. Praeger), Journal of Combinatorial Designs 24 (2016) 151-164. Article.

Applied differential geometry

1. Elastica in SO(3) (with L. Noakes), Journal of the Australian Mathematical Society 83 (2007) 105-125. Article.
2. Geometry for robot path planning (with L. Noakes), Robotica 25 (2007) 691-701. Article.
3. Bézier curves and C² interpolation in Riemannian manifolds (with L. Noakes), Journal of Approximation Theory 148 (2007) 111-127. Article.
4. Higher order geodesics in Lie groups, Mathematics of Control, Signals, and Systems 19 (2007) 235-253. Article.
5. On parametric smoothness of generalised B-spline curves, Computer Aided Geometric Design 23 (2006) 655-668. Article.
6. Quadratures and cubics in SO(3) and SO(1, 2) (with L. Noakes), IMA Journal of Mathematical Control and Information 23 (2006) 463-473. Article.
7. C² spherical Bézier splines (with L. Noakes), Computer Aided Geometric Design 23 (2006) 261-275. Article.
8. Null Riemannian cubics in tension in SO(3) (with L. Noakes), IMA Journal of Mathematical Control and Information 22 (2005) 477-488. Article.

Errata and addenda

• M, B and Co₁ are recognisable by their prime graphs. On p. 195, the definition V = K/L should be replaced by V = L/K. On p. 196, the sentence "Note that there are exactly 8 conjugacy classes of elements of odd order in Co₁ that intersect both Co₃ and H." should be replaced by "Note that there are exactly 8 conjugacy classes of elements of odd order in H that belong to Co₁-classes that also intersect Co₃.".
• Generalised polygons admitting a point-primitive almost simple group of Suzuki or Ree type. There is a missing case in Section 5 owing to a previously unknown maximal subgroup PGL₂(13) of ²F₄(8) discovered by David Craven (see Remark 4.11 here). Here is a proof of our Theorem 1.1 in this case (using the notation of our Section 5). If S=²F₄(8) and G_x=PGL₂(13) then |S:G_x| = 121,293,659,660,982,681,600 = 2³³k with k odd. If |P| is divisible by 2³³, then our Lemma 2(i) implies that |P| > 2⁹⁹. Since |S:G_x| < 2⁹⁹, it cannot be the case that |P| = |S:G_x|, so S cannot act primitively on P with point stabiliser G_x.
• Algorithms to identify abundant p-singular elements in finite classical groups. In Definition 1.1(ii), both instances of ζ^-1 should read ζ^-q.
• On proportions of pre-involutions in finite classical groups. In Table 1, the 1/4 in the last line can be replaced by 1/2. In Theorem 1.5(iii) and Corollary 1.6(ii), the right-hand sides of the inequalities can be multiplied by 2. (The results are correct as stated in the paper, but can be improved as described here.)