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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.


Volltext §
DOI: 10.5445/IR/1000117988
Veröffentlicht am 02.04.2020
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Algebra und Geometrie (IAG)
Publikationstyp Hochschulschrift
Publikationsdatum 02.04.2020
Sprache Englisch
Identifikator KITopen-ID: 1000117988
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 80 S.
Art der Arbeit Dissertation
Fakultät Fakultät für Mathematik (MATH)
Institut Institut für Algebra und Geometrie (IAG)
Prüfungsdatum 17.03.2020
Schlagwörter Inzidenzgeometrie, Design, affines Unital, Automorphismus, Parallelismus, Translation
Referent/Betreuer Herrlich, F.
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