Three Affine SL(2,8)-Unitals

Parallelisms and Translations of (Affine) SL(2,q)-Unitals

Unitals can be obtained as closures of affine unitals via parallelisms. The isomorphism type of the closure depends on the chosen parallelism, which need not be unique. For affine $\operatorname{SL}(2,q)$-unitals, we introduce a class of parallelisms for odd order and one for square order. Translations are automorphisms of unitals, fixing each block through a given center. For each of the known parallelisms of affine $\operatorname{SL}(2,q)$-unitals, we compute all possible translations with centers on the block at infinity.

Automorphisms of (Affine) SL(2,q)-Unitals

$\operatorname{SL}(2,q)$-unitals are unitals of order $q$ admitting a regular action of $\operatorname{SL}(2,q)$ on the complement of some block. They can be obtained from affine $\operatorname{SL}(2,q)$-unitals via parallelisms. We compute a sharp upper bound for automorphism groups of affine $\operatorname{SL}(2,q)$-unitals and show that exactly two parallelisms are fixed by all automorphisms. In $\operatorname{SL}(2,q)$-unitals obtained as closures of affine $\operatorname{SL}(2,q)$-unitals via those two parallelisms, we show that there is one block fixed under the full automorphism group.

奇数阶Buekenhout-Metz酉极性中的三重O'Nan构型

Triple O'Nan Configurations in Buekenhout-Metz Unitals of Odd Order

该研究拓展了Feng和Li的相关工作，证明了奇数阶Buekenhout-Metz酉极性中存在由三个不同O'Nan构型组成的三重O'Nan构型。

二阶广义六边形中的一个 unital，以及有限李型群之间的一个例外同构

A unital in the generalized hexagon of order two, and an exceptional isomorphism between finite groups of Lie type

本文在二阶域上的八元数代数中构造三阶 Hermitian unital 模型，以说明例外李型有限群与 Hermite 型半相似群之间的已知同构。

A classification of one dimensional affine rank three graphs

The rank three subgroups of a one-dimensional affine group over a finite field were classified in 1978 by Foulser and Kallaher. Although one can use their results for a classification of corresponding rank three graphs, the author did not find such a classification in a literature. The goal of this note is to present such a classification. It turned out that graph classification is much simpler than the group one. More precisely, it is shown that the graphs in the title are either the Paley graphs or one of the graphs constructed by Van Lint and Schrijver or by Peisert. Our approach is based o…

Factorization length distribution for affine semigroups IV: a geometric approach to weighted factorization lengths in three-generator numerical semigroups

For numerical semigroups with three generators, we study the asymptotic behavior of weighted factorization lengths, that is, linear functionals of the coefficients in the factorizations of semigroup elements. This work generalizes many previous results, provides more natural and intuitive proofs, and yields a completely explicit error bound.