# Algorithmic analysis of presentations of groups and modules

Aachen / Publikationsserver der RWTH Aachen University (2009) [Dissertation / PhD Thesis]

Page(s): 182 S.

Abstract

In this thesis Janet's algorithm for computing bases in polynomial rings has been applied to two algebraic problems:1. the explicite basis construction in the context of the Quillen-Suslin Theorem,2. the analysis of finite group presentations, more precisely enumerating finite L_2-quotients of finitely presented groups.The emphasis has been put on the constructive aspects of all considered issues and on the development of methods and algorithms. The thesis is accompanied by two Maple-packages QuillenSuslin and PSL. The first chapter is devoted to algorithmic and computational issues concerning polynomial systems with integer coefficients such as the construction of a maximal ideal containing a given ideal or the computation of the set of all minimal associated prime ideals. In Chapter 2 an algorithmic proof of the Quillen-Suslin Theorem is given, in particular an algorithm for computing a basis of free module over a polynomial ring over principle ideal domains. The problem is represented in the language of completing unimodular matrices to square invertible matrices over a polynomial ring. The general inductive algorithm has been improved by the use of some heuristic methods. Since no working implementation carrying out the computation of a basis for a free module was available, the presented methods have been implemented in the Maple-package QuillenSuslin. At the end a few applications of the Quillen-Suslin theorem, more precisely of the actual basis computation for a free module, to system theory and to algebraic geometry are presented to demonstrate the power of the implementation. Finally, a very rich library of examples illustrating the use and applications of the QuillenSuslin package is given in Appendix B. Chapter 3 deals with the analysis of finitely presented groups. Among many questions one can ask about a given finitely presented group, the algorithmic decision whether G is finite or not, stands out as being impossible in general. This thesis gives an algorithm, which can decide (in case the number of generators is smaller than 4) whether the number of normal subgroups N of G with the quotient G/N of type L_2 is finite or infinite and is therefore called a L_2-quotient algorithm. A finite group is called to be of L_2-type, if it is isomorphic to PSL (2,p^n) or to PGL (2,p^n) for some prime number p and some natural number n. The significance of the algorithm is that for the first time one can enumerate all factor groups in a big class of finite simple groups. In case the computation yields infinitely many normal subgroups of type L_2, one has a particularly strong form of an infiniteness proof. If infinitely many primes are involved one can even exhibit a non soluble infinite matrix group of degree 2 over a field of characteristic zero as epimorphic image. In general one can even distinguish certain kinds of infiniteness by using the notion Krull dimension for commutative domains

## Authors

### Authors

Fabianska, Anna Wiktoria