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Markus J. Stroppel

Publications and source records attributed to Markus J. Stroppel.

8 recordsLinked to original sources
arXiv2025

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

We construct a model of the Hermitian unital of order 3 (obtained from the non-degenerate hermitian form in three variables over the field of order 9) inside the octonion algebra over the field of order 2. This construction is invariant under the automorphism group of that algebra, and explains the known isomorphism from the finite group of exceptional Lie type onto the group of semi-similitudes of the hermitian form.

math.GR↗
arXiv2025

Confluence Graphs of Unitals

We show that the cliques of maximal size in the confluence graph of an arbitrary unital of order $q>2$ have size $q^2$, and that these cliques are the pencils of all blocks through a given point. This solves the Erdős-Ko-Rado problem for all unitals. We also determine all maximal cliques of the confluence graph of the Hermitian unitals. As an application, we show that the confluence graph of an arbitrary unital unambiguously determines the unital. Along the way, we show that each linear space with $q^2$ points such that the sizes of both point rows and line pencils are bounded above by $q+1$ embeds in a projective plane of order $q$.

math.CO↗
arXiv2023

Subalgebras of Octonion Algebras

For an arbitrary octonion algebra, we determine all subalgebras. It turns out that every subalgebra of dimension less than four is associative, while every subalgebra of dimension greater than four is not associative. In any split octonion algebra, there are both associative and non-associative subalgebras of dimension four. Except for one-dimensional subalgebras spanned by idempotents, any two isomorphic subalgebras are in the same orbit under automorphisms.

math.RA↗
arXiv2022

Hodge Operators and Exceptional Isomorphisms between Unitary Groups

We give a generalization of the Hodge operator to spaces $(V,h)$ endowed with a hermitian or symmetric bilinear form $h$ over arbitrary fields, including the characteristic two case. Suitable exterior powers of $V$ become free modules over an algebra $K$ defined using such an operator. This leads to several exceptional homomorphisms from unitary groups (with respect to $h$) into groups of semi-similitudes with respect to a suitable form over some subfield of $K$. The algebra $K$ depends on $h$; it is a composition algebra unless $h$ is symmetric and the characteristic is two.

math.GR↗
arXiv2022

Finite subunitals of the Hermitian unitals

Every finite subunital of any generalized hermitian unital is itself a hermitian unital; the embedding is given by an embedding of quadratic field extensions. In particular, a generalized hermitian unital with a finite subunital is a hermitian one (i.e., it originates from a separable field extension).

math.CO↗
arXiv2020

Moufang sets generated by translations in unitals

We consider unitals of order $q$ with two points which are centers of translation groups of order $q$. The group $G$ generated by these translations induces a Moufang set on the block joining the two points. We show that $G$ is either $\operatorname{SL}(2,\mathbb{F}_q)$ (as in all classical unitals and also in some non-classical examples), or $\operatorname{PSL}(2,\mathbb{F}_q)$, or a Suzuki or a Ree group. Moreover, $G$ is semi-regular outside the special block.

math.GR↗