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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.

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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 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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