#### Title

On Flips of Unitary Buildings

#### Date of Award

2010

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Department

Mathematics

#### First Advisor

Dr. Rieuwert Blok, PhD

#### Abstract

In the Gorenstein-Lyons-Solomon revision of the proof of the classification theorem for finite simple groups, one of the techniques in the identification step involves identifying a particular finite simple group (the minimal counterexample to the theorem) with a known finite simple group. Phan's theorems and the Curtis-Tits theorem provide some techniques for accomplishing this identification. Both these theorems can be phrased in terms of groups acting flag transitively on a simply connected geometry. One method of producing geometries suitable for proving Phan-type theorems is to study geometries induced by flips on twin buildings.

The purpose of this work is to classify flips of the building associated to the geometry of totally isotropic subspaces of a non-degenerate unitary spaces over a finite field of odd characteristic. A secondary goal is to study some properties of geometries related to these flips. We prove that there are up to similarity only four flips, and that for sufficiently large unitary spaces the resulting geometries are simply connected. We then appeal to Tits' Lemma to prove Phan-type theorems for certain flag-transitive automorphism groups of these geometries.

#### Recommended Citation

Carr, Benjamin, "On Flips of Unitary Buildings" (2010). *Mathematics Ph.D. Dissertations*. 5.

https://scholarworks.bgsu.edu/math_diss/5