Abstract
Unitals can be obtained as closures of affine unitals via parallelisms. Affine \({{\,\mathrm{SL}\,}}(2,q)\)-unitals are affine unitals of order q admitting a regular action of \({{\,\mathrm{SL}\,}}(2,q)\). The construction of those affine unitals is due to Grundhöfer, Stroppel and Van Maldeghem and motivated by the action of \({{\,\mathrm{SL}\,}}(2,q)\) on the classical (Hermitian) unital. For affine \({{\,\mathrm{SL}\,}}(2,q)\)-unitals, we introduce a class of parallelisms for odd order and one for square order and compute their stabilizers. For each of the known parallelisms of affine \({{\,\mathrm{SL}\,}}(2,q)\)-unitals, we compute all translations with centers on the block at infinity.
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Notes
In [7], the parallelism \(\pi _{\mathsf {odd}}\) is called \(\pi ^\square \) and \(\pi _{\mathsf {odd}}'\) is called \({}^{\square }\pi \).
These parallelisms for order 4 were found during the Leonid meteor shower in November 2018. Since they resulted in twelve new unitals at once, the author was reminded of a unital shower and hence called the twelve unitals \(\mathbb {U}_1, \ldots ,\mathbb {U}_{12}\) the Leonids unitals in her PhD thesis [7].
References
GAP—Groups, Algorithms, and Programming, Version 4.8.10. The GAP Group (2018). https://www.gap-system.org
Grundhöfer, T., Stroppel, M.J., Van Maldeghem, H.: Moufang sets generated by translations in unitals (2020). arXiv: 2008.11445 [math.GR]
Grundhöfer, T., Stroppel, M.J., Van Maldeghem, H.: A nonclassical unital of order four with many translations. Discrete Math. 339(12), 2987–2993 (2016)
Grundhöfer, T., Stroppel, M.J., Van Maldeghem, H.: Unitals admitting all translations. J. Combin. Des. 21(10), 419–431 (2013)
Mezõfi, D., Nagy, G.P.: New Steiner systems from old ones by paramodifications (2020). arXiv: 2003.09233 [math.CO]
Möhler, V.: Automorphisms of (Affine) SL(2,q)-Unitals (2020). arXiv: 2012.10116 [math.CO]
Möhler, V.: SL(2,q)-Unitals. Ph.D. Thesis. Karlsruher Institut für Technologie (KIT) (2020)
Möhler, V.: Three Affine SL(2,8)-Unitals. Beitr. Algebra Geom. (to appear 2022). arXiv: 2012.10134 [math.CO]
Nagy, G.P., Mezõfi, D.: UnitalSZ, algorithms and libraries of abstract unitals and their embeddings, Version 0.5. GAP package (2018). https://nagygp.github.io/UnitalSZ/
Robinson, D.J.S.: A Course in the Theory of Groups. Graduate Texts in Mathematics, vol. 80, 2nd edn Springer, New York (1996)
Acknowledgements
The author wishes to warmly thank her thesis advisor Markus J. Stroppel for his highly valuable support in each phase of this research.
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Most of the results in the present paper have been obtained in the author’s Ph. D. thesis [7], where detailed arguments can be found for some statements that we leave to the reader here.
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Möhler, V. Parallelisms and translations of (affine) \(\hbox {SL}(2,q)\)-unitals. J. Geom. 112, 44 (2021). https://doi.org/10.1007/s00022-021-00611-5
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DOI: https://doi.org/10.1007/s00022-021-00611-5