Algebra Seminar

Characterizing a group in terms of a finite set of generators and relations between them is a technique that is used in several situations, not only in group theory itself, but also in geometry, graph theory and even in quantum physics. Therefore it exists a constant interest in dealing with finitely presented groups.

Though a finite presentation is usually a very compact description of a group, it is very difficult to analyze its structure or even to do concrete calculations in the group. Algorithms like the Todd Coxeter algorithm may solve these problems in many examples, but its success depends not only on the group, but crucially on the individual presentation. Often massive computer power is also needed to come to a solution.

Another approach for analyzing the structure of a finitely presented group G is to find quotients of G, i.e. epimorphic images of G onto some 'well-known' groups. If the quotient is suitably large, a lot of calculations can be done in the quotient instead of in G itself, and many questions about G can be answered.

The first of these algorithms is the abelian quotient algorithm. It calculates an epimorphism onto an abelian group and characterizes the commutator factor G/G'. In the last few years algorithms for constructing p-group quotients (E. O'Brian) nilpotent group quotients (W. Nickel) have been developed and used successfully for large quotients. These lead to the determination of the lower p-central series resp. the lower central series of G.

Since the commutator series is an interesting structural invariant of a group, the question for soluble quotients appears to be a natural extension.

In this talk I will outline a finite soluble quotient algorithm and its background in the theory of soluble groups, representation theory and group cohomology.