Schur Multiplier - Relation To Efficient Presentations

Relation To Efficient Presentations

In combinatorial group theory, a group often originates from a presentation. One important theme in this area of mathematics is to study presentations with as few relations as possible, such as one relator groups like Baumslag-Solitar groups. These groups are infinite groups with two generators and one relation, and an old result of Schreier shows that in any presentation with more generators than relations, the resulting group is infinite. The borderline case is thus quite interesting: finite groups with the same number of generators as relations are said to have a deficiency zero. For a group to have deficiency zero, the group must have a trivial Schur multiplier because the minimum number of generators of the Schur multiplier is always less than or equal to the difference between the number of relations and the number of generators, which is the negative deficiency. An efficient group is one where the Schur multiplier requires this number of generators.

A fairly recent topic of research is to find efficient presentations for all finite simple groups with trivial Schur multipliers. Such presentations are in some sense nice because they are usually short, but they are difficult to find and to work with because they are ill-suited to standard methods such as coset enumeration.