Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Claus Fieker
University of Sydney
Working in the multiplicative group of a number field
The multiplicative group of a number field is rather large and difficult to work in on a computer: we clearly
do not have a finitely generated Z-module structure that can be used. Therefore to use the multiplicative
group in applications, one frequently starts by creating a finitely generated sub-group that is large enough
to contain a solution but small enough to allow effective manipulation. Apart from the finite generation,
a second problem comes from the (neccessary) use of logarithms to linearize the structure, it implies
that the linear structure is only approximated and not exactly represented. In this context there are a few
important problems to solve:
- given a finite number of non-zero number field elements, can we compute the Z-module structure?
- given a tentative sub-group, can we enlarge it systematically?
- given a particular problem, can we find an effective set of generators for the part that we are interested
in?
- given an element in the finitely generated group, can we find nicer representatives?
Examples here are the computation of the class group, S-unit group, solution of norm equations, splitting
of co-cycles in cohomology groups and p-Selmer group computations. I will indicate algorithmic solutions
to some of the problems, classical solutions as well as new ones based on p-adic techniques.
Max Horn
Technische Universität Braunschweig
Polyzyklische Quotienten und nicht-kommutative Gröbner-Basen
Wir beschreiben, wie man polyzyklische Quotienten von (endlich) präsentierten Gruppen algorithmisch
berechnen kann, und stellen unsere Implementierung in GAP vor. Zentral dafür sind Methoden, um
verallgemeinerte Gröbner-Basen in integralen Gruppenringen von polyzyklischen Gruppen zu berechnen;
diese Ringe sind dabei im allgemeinen nicht kommutativ. Zum Ende geben wir ein paar Beispiele und
weitere Anwendungen dieser Gröbnerbasenmethoden an.
Sebastian Jambor
RWTH Aachen
An L3-U3-quotient algorithm
Given a finitely presented group G on two generators, the L3-U3-quotient algorithm enumerates all normal
subgroups N � G such that G/N is isomorphic to PSL(3,q) or PSU(3,q). This is done simultaneously
for any q, and even works if G has infinitely many factor groups isomorphic to PSL(3,q) or PSU(3,q) (in
particular, this gives a proof that G is infinite, in this case).
The algorithm uses methods from representation theory and from commutative algebra. In the talk, I will
present some of the ideas of the algorithm and give a demonstration with several examples.
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