# An L 3 -U 3 -quotient algorithm for finitely presented groups

• Ein L3-U3-Quotientenalgorithmus für endlich präsentierte Gruppen

Jambor, Sebastian; Plesken, Wilhelm (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2012)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2012

Abstract

The thesis describes the development of an L3-U3-quotient algorithm for finitely presented groups on two generators, which finds all factor groups of a finitely presented group which are isomorphic to one of the groups PSL(3, q), PSU(3, q), PGL(3, q), or PGU(3, q). Here q is an arbitrary prime power which is not part of the input of the algorithm, so the algorithm finds all possible choices of q by itself. The motivation for such an algorithm is to study and understand finitely presented groups. Results from the middle of the last century show that central questions concerning these groups are undecidable in general. The most famous is the word-problem: there cannot exist an algorithm which decides the equality of two elements of a finitely presented group. Nevertheless, algorithms have been developed which give structural information about finitely presented groups. One class of such algorithms are the quotient algorithms, which find all factor groups of a given finitely presented group that have a certain structure. Until a few years ago, all of those algorithms only worked for a finite set of factor groups or for soluble groups. In 2009, Plesken and Fabia'nska developed the first algorithm which can compute all factor groups in an infinite class of non-soluble groups. It determines all factor groups of a finitely presented group G which are isomorphic to one of the groups PSL(2, q) or PGL(2,q), simultaneously for any prime power q. The present thesis is a continuation of those ideas. For the formulation of the algorithm, various results from representation theory and from commutative algebra are needed. These are stated and proved in this work. The character of a representation has been an important tool in ordinary representation theory, i.e., of representations of finite groups over fields of characteristic zero. The results in this thesis show that it is still an invaluable tool for representations of arbitrary groups over arbitrary fields. A method in commutative algebra is the determination of all minimal associated prime ideals of a given ideal. This dissertation presents an algorithm which improves on the runtime of existing algorithms for important examples. The results in representation theory and in commutative algebra are applied to the L3-U3-quotient algorithm, but they are of general interest as well. The L3-U3-quotient algorithm is implemented in the computer algebra system Magma. This implementation is applied to several examples of finitely presented groups. Furthermore, results of this thesis are used to prove generalizations of theorems of P. Hall and Lubotzky.