The course is supposed to give a gentle introduction to group theory from a computational point of view. That is not only the mathematical theory will be presented, but also the computational aspects considered and it is actually the latter that will determine the course of the lecture. We will do a broad overview of finite permutation groups (i.e., groups generated by a set of permutations), describing specific algorithms for computations inside them. We will also touch a few aspects in the direction of their classification. The main aim of this part will be to become acquainted with practical computability in groups, i.e., to answer broadly what kind of problems in groups could be solved in practice by computational methods.
The second part of the lecture will be focused on finitely presented groups. These are potentially infinite groups defined by a set of generators together with equations defining the relations among them. We will describe how such groups could be effectively implemented on a computer and we will cover algorithms for finding their quotients, presentations of their subgroups, etc. We will then move to the topics of formal languages used to describe a procedure known as the Knuth-Bendix completion which could be used (in specific circumstances) to solve the word problem. Finally we will use automata theory to decide the finiteness of a finitely presented group.
In exercise sessions you will be asked to produce working implementations of the data structures and the algorithms covered during the lectures. The intention is to deepen the understanding of the material by hands-on approach and to encourage learning by experimenting in small groups working together.
The exercises will be run using Julia programming language.
While no formal knowledge of programming is required an affinity to programming and a general idea of how computers work are goiong to make the course much more enjoyable for you.