Group generation: where algebra, geometry and computation meet

Abstract: A minimal generating set of a group is a set of elements that generates the group such that, if any element is removed, the remaining set no longer generates the group. For example, the symmetric group of degree n has minimal generating sets whose sizes range from 2 up to n−1. The study of minimal generating sets is a topic of particular interest because of its connections with two important areas: discrete geometry and computational algebra. In discrete geometry, there is a well-known one-to-one correspondence between certain minimal generating sets of involutions satisfying specific axioms and regular polytopes. In computational algebra, the connectivity of product replacement graphs, an important tool used in algebra systems to generate random elements of groups, depends on the parameters m(G) and d(G), which denote the maximal and minimal sizes of minimal generating sets of a group G, respectively.

In this seminar, I will give an overview of the topic and discuss some of its main results and applications.