Über die Elementarteiler der Inzidenzmatrix des Ree-Unitals

• On the elementary divisors of the incidence matrix of the Ree unital

Ringe, Christian; Hiß, Gerhard (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2008)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2008

Abstract

The Ree unital is a certain 2-design which admits the Ree group G:=R(q) of parameter q (where q is a power of 3 with odd exponent) as a group of design-automorphisms. We investigate the elementary divisors of the incidence matrix of the Ree unital. The elementary divisors mentioned above reflect the structure of a finite abelian group. In order to determine the structure of this group, we apply some methods of module theory and of modular representation theory. A complete determination of the structure fails in the general case. However, we can reduce the number of possibilities for the 2-component to three. In certain cases we succeed in completely determining the components of odd order. Another part of this thesis deals with a function field F over Fq (where Fq is the finite field with q elements), which admits the Ree group G as full Fq-automorphism group. We succeed in proving that we can identify the Fq-rational places of F with the points of the Ree unital, and that the set of Fq-rational places with the operation of G provides a model of the Ree unital. We investigate the group D0 of divisor classes of degree 0 of F and some of its subgroups related to the Ree unital. For this purpose we also apply some methods of modular representation theory. A complete determination of the structure of D0 fails. However, we can determine the components of odd order of D0 in some cases. In case of the smallest Ree unital, that is in case q=3, we can completely answer all these questions. In order to do this, we apply some results of computer calculations. The last chapter of this dissertation deals with certain 3-designs, the Moebius planes (also known as inversive planes). We apply some methods of preceding chapters of this thesis to this situation. In all the investigated cases we succeed in completely determining the elementary divisors of the respective incidence matrix.