Left 3-Engel elements in groups

The project falls within an area of pure mathematics that deals with the notion of symmetry in a very general sense. To describe a given symmetry one associates to it an algebraic system, a group, consisting of the operations that preserve the given symmetry. In this way one is able to study and analyse the symmetry with mathematical rigour focusing on the algebraic system that captures it. Normally the collection of operations is generated by a subset. For example for the Rubics cube all the operations are generated by 6 basic operations corresponding to the 6 sides of the cube. The total number of operations is however enormous which is the reason why the Rubics cube is tricky to solve. When the group can be described in terms of finitely many operations we say that it is finitely generated. A central question regarding such finitely generated groups is whether it must be finite if all the operations are of finite order (i.e. if, for each operation, if it is repeated a finite number of times we are back where we started).

The precise mathematical questions were formulated by the English mathematician William Burnside in 1902. The answer to the general question turned out to be negative and as a result a large main subbranch of group theory opened up that remains one of the central branches of group theory today with a number of major challanges as well as breakthroughs of which one of the most spectacular is the `solution to the restricted Burnside problem' by the Russian mathematician Efim Zel'manov for which he received the Fields medal in 1994. The work of Zel'manov has had a profound impact on Group Theory in the last two decades.

Engel conditions are a certain type of technical algebraic conditions that appear in the study of these problems and are crucial in both gaining better understanding of the Burnside problems as well as being of interest in their own right when studying symmetry. The central problem of the proposed project is a well known difficult open problem in the area and involves a systematic study of finitely generated groups with specific properties. The question is whether these groups have a certain finite-like structure (called nilpotent) and the conclusion in that case would be in particular that if the generators are of finite order then the group would be finite.

The approach will be based on a recent advance made by the principal investigator where the problem has been solved when the number of generators is three as well as for any number of generators when the group has exponent 5 (meaning that all the non-trivial operations have order 5). The aim would be to use the techniques developed in this work to tackle the more general problem.

The project concerns a fundamental problem in group theory. There would be an immediate strong impact on the community of mathematicians working within the large area of Burnside type questions and Engel like conditions in group theory. The central problem of the proposal is however a longstanding question about Engel conditions in group theory and any major breakthrough would also potentially have wider impact on group theory and other mathematical theories.

Mathematics lies at the very foundation of scientific knowledge although the underlying mathematical theories are usually related to the actual applications in indirect ways. It can take a long time for a given mathematical theory to find applications outside the sphere of mathematics. Group theory, the mathematical theory that handles symmetry, originates with the ingenious work of the French mathematician Evariste Galois (1811-1832) in his work on polynomial equations. The work involves a systematic study of the underlying symmetry that is present. This theory has however since then found wide applications both within other branches of mathematics as well as outside mathematics where symmetry plays a large role including Chemistry, Particle Physics and Cryptography. A fundamental question in group theory is in fact a fundamental question about symmetry, and a significant advance in our understanding of symmetry increases the potential of group theory to have wide applications.

Bringing together strong mathematicians in the field through setting up a miniconference in Bath as well as participating in conferences would accelerate impact and any major breakthrough would have a longterm effect both on this research area and potentially wider as the the questions are fundamental in understanding symmetry. The subbranch is studied by a wide range of mathematicans with leading figures in UK and around the world.

The project will also provide a training and a platform for a postdoctoral researcher in the area to develop his or her skills and knowledge. Part of the research grant would be used to finance a visit of the PDRA to another research centre that the PI has a link with. This would widen the expertise of the PDRA as well as providing him or her with links with some major figures in the area. The project involves some lengthy and complicated computations and some training in using the group theory software package GAP would probably be required as well.