# SL(2,q)-Unitals

Möhler, Verena

##### Abstract (englisch):
Unitals of order $n$ are incidence structures consisting of $n^3+1$ points such that each block is incident with $n+1$ points and such that there are unique joining blocks. In the language of designs, a unital of order $n$ is a $2$-$(n^3+1,n+1,1)$ design. An affine unital is obtained from a unital by removing one block and all the points on it. A unital can be obtained from an affine unital via a parallelism on the short blocks. We study so-called (affine) $\mathrm{SL}(2,q)$-unitals, a special construction of (affine) unitals of order $q$ where $q$ is a prime power. We show several results on automorphism groups and translations of those unitals, including a proof that one block is fixed by the full automorphism group under certain conditions. We introduce a new class of parallelisms, occurring in every affine $\mathrm{SL}(2,q)$-unital of odd order. Finally, we present the results of a computer search, including three new affine $\mathrm{SL}(2,8)$-unitals and twelve new $\mathrm{SL}(2,4)$-unitals.

 Zugehörige Institution(en) am KIT Institut für Algebra und Geometrie (IAG) Publikationstyp Hochschulschrift Publikationsdatum 02.04.2020 Sprache Englisch Identifikator KITopen-ID: 1000117988 Verlag Karlsruhe Umfang 80 S. Art der Arbeit Dissertation Fakultät Fakultät für Mathematik (MATH) Institut Institut für Algebra und Geometrie (IAG) Prüfungsdatum 17.03.2020 Referent/Betreuer Prof. F. Herrlich Schlagwörter Inzidenzgeometrie, Design, affines Unital, Automorphismus, Parallelismus, Translation
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