# Left 3-Engel elements in groups

Lead Research Organisation:
University of Bath

Department Name: Mathematical Sciences

### Abstract

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 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.

### Planned Impact

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.

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.

### Organisations

## People |
## ORCID iD |

Gunnar Traustason (Principal Investigator) |

### Publications

Hadjievangelou A
(2021)

*Left 3-Engel elements in locally finite 2-groups*in Communications in Algebra
Hadjievangelou A
(2020)

*Locally finite p-groups with a left 3-Engel element whose normal closure is not nilpotent*in International Journal of Algebra and Computation
Jabara E
(2019)

*Left $3$-Engel elements of odd order in groups*in Proceedings of the American Mathematical Society
Noce M
(2020)

*A left 3-Engel element whose normal closure is not nilpotent*in Journal of Pure and Applied Algebra
Tracey G
(2018)

*Left 3-Engel elements in groups of exponent 60*in International Journal of Algebra and ComputationDescription | 1. The central question of the project is to settle the longstanding question whether, in any given group G, a left 3-Engel element a must be contained in the locally nilpotent radical. In a joint paper with Enrico Jabara, `Left 3-Engel elements of odd order in groups', we show that this is the case when the order of a is an odd number. This goes a long way towards solving the central question. The approach is to work with the sandwich groups but these are the analogues to sandwich algebras that were introduced by Kostrikin and Zel'manov in the 80's. One of the main ingredients for the solution of the `Restricted Burnside Problem' is a Theorem of Kostrikin and Zel'manov from 1988, but they show that any finitely generated sandwich algebra is nilpotent. With E. Jabara, we show that every finitely generated sandwich group, generated by elements of odd order, is nilpotent. 2. The question remains if one can extend this result to any left 3-Engel element a of finite order. With my PDRA, Gareth Tracey, we reduced the problem to the case when a is of order 2. We also prove some partial results in this direction in our paper `Left 3-Engel elements in groups of exponent 60' where we in particular show that a left 3-Engel element of order 2 is in the locally nilpotent radical if furthermore the normal closure of a in G has not elements of order 8. 3. Another natural question that arises is whether the normal closure of a given left 3-Engel element is not only locally nilpotent but globally nilpotent. In a paper with Marialaura Noce and Gareth Tracey we showed that this does not have to be the case by giving an example of locally finite 2-group containing a left 3-Engel element whose normal closure is not nilpotent. In a joint work with Marialaura Noce and my research student Anastasia Hadjivangelou we extended this to locally finite p-groups where p is odd and this work has now appeared in `International Journal of Algebra and Computation'. Another joint work with Anastasia that generalises the construction we came up with for locally finite 2-groups has been published in `Communications in Algebra'. There are now two and a half months that remain of this challenging project. I am pleased to see that essentially the original objectives have been met. In fact the progress has been far beyond my expectations. |

Exploitation Route | These findings are important steps towards settling the general question, whether a left 3-Engel element is always contained in the locally nilpotent radical. Having settled this for the case when the left-Engel element is of odd order we feel that we are close to a complete solution. The hope is that we or some of our colleagues may be able to accomplish this using the breakthrough findings. The findings are also interesting in their own right. There are connections to groups of prime power exponent and in the paper with Enrico Jabara, we apply the main result to obtain some finiteness criteria for the variety of groups of exponent 9. The results may find further applications in similar vain. |

Sectors | Other |

Description | Eloisa Detomi and Andrea Lucchini Padova |

Organisation | University of Padova |

Country | Italy |

Sector | Academic/University |

PI Contribution | I have contributed to a joint project `Groups satisfying a strong complement property'. I visited the University of Padova in July 2018 in order to work on this project. My PDRA, Gareth Tracey, also visited Padova in October 2017 for a week in October 2017 and has been collaborating with Andrea Lucchini on two projects. |

Collaborator Contribution | Andrea and Eloisa contributed to the first project and we have published a joint paper `Groups satisfying a strong complement property' (see below). Andrea collaborated with Gareth Tracey on two projects. For more details see below. |

Impact | `Groups satisfying a strong complement property' ( E. Detomi, A. Lucchini, M. Moscatiello, P. Spiga, G. Traustason), J. Algebra 535 (2019), 35-52. `Finite groups with large Chebotarev invariant` (A. Lucchini, G. Tracey), Israel J. Math. 235 (2020), no. 1, 169-182. `Generating maximal subgroups of almost simple groups (with A. Lucchini and C. Marion), Forum Math. Sigma 8 (2020), Paper No. e32, 67pp. |

Start Year | 2017 |

Description | Enrico Jabara. Co-author of the publication `Left 3-Engel elements of odd order in groups' |

Organisation | Ca' Foscari University of Venice |

Country | Italy |

Sector | Academic/University |

PI Contribution | I collaborated with Enrico on the paper `Left 3-Engel elements of odd order in groups' that is central to the project on Left 3-Engel elements. We have also written another joint paper in 2020 on (n+1/2)-Engel groups. |

Collaborator Contribution | Enrico is my co-author of the paper `Left 3-Engel elements of odd order in groups' and `On (n+1/2)-Engel groups'. |

Impact | E. Jabara and G. Traustason, Left 3-Engel elements of odd order in groups, Proc. Am. Math. Soc. , 147 NO. 5 (2019), 1921-1927. E. Jabara and G. Traustason, On (n+1/2)-Engel groups. J. Group Theory 23 (2020), no. 3, 503-515. |

Start Year | 2018 |

Description | Evgeny Khukhro, Linclon, UK. A research collaborator. |

Organisation | University of Lincoln |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | I have published a paper with him and Pavel Shumyatsky. |

Collaborator Contribution | He worked with me and Pavel Shumyatsky on a paper (see below). He is also participated in a conference that we was held in Bath, 24-26 April 2019. |

Impact | E. Khukhro, P. Shumyatsky and G. Traustason, Right Engel-Type subgroups and length parameters of finite groups, J. Aust. Math. Soc. 109 (2020), no. 3, 340-350. |

Start Year | 2017 |

Description | Gustavo Fernandez-Alcober. Research collaborator. |

Organisation | University of the Basque Country |

Country | Spain |

Sector | Academic/University |

PI Contribution | My PDRA, Gareth Tracey, collaborated with Gustavo and his student Marialaura on a research project on Branch Groups. Gareth visited Bilbao in April 2019 to work on this project. In 2020 Marialaura replaced Gareth as my PDRA and worked with me on the central project on Left 3-Engel elements in groups, resulting in two papers, one joint with me and Gareth that has been published and another joint with me and my PhD student Anastasia Hadjievangelou that has been submitted few months ago. |

Collaborator Contribution | Gustavo and his student Marialaura Noce have been collaborating with my PDRA and me on a couple of research projects and visited us in Bath for a week in February 2019 in order to work with us. |

Impact | Engel elements in weakly branch groups (G, Fernandez-Alcober, M. Noce, G. Tracey), J. Algebra 554 (2020), 54-77. Locally finite p-groups with a left 3-Engel element whose normal closure is not nilpotent (A. Hadjievangelou, M. Noce, G. Traustason) , International Journal of Algebra and Computation, to appear. |

Start Year | 2018 |

Description | Patrizia Longobardi and Mercede Maj University of Salerno |

Organisation | University of Salerno |

Country | Italy |

Sector | Academic/University |

PI Contribution | I visited Salerno for two weeks in November-December 2019 to work on a joint project with Pavel Shumyatsky (University of Brasila). This led to a joint publication. |

Collaborator Contribution | Collaborated on the joint project described above. |

Impact | P. Longobardi, M. Maj, P. Shumyatsky, G. Traustason. Groups with boundedly many commutators of maximal order. J. Algebra 657 (2021), 269-282. |

Start Year | 2019 |

Description | Pavel Shumyatsky. Research collaborator. |

Organisation | University of Brasilia |

Country | Brazil |

Sector | Academic/University |

PI Contribution | I visited Pavel Shumyatsky in Brasilia for two weeks in November 2017 and again for a week in February 2019 where I also participated in the `XI Summer Workshop in Mathematics, Department of Mathematics, Universidade de Braslilia, February 18-22 2019. We have written a paper together in collaboration with Evgeny Khukhro, Lincoln, that was published in 2020. He visited me again in April 2019 that led to a collaboration with Patrizia Longobardia and Mercede Maj in Salerno and a joint publication to appear in 2021. |

Collaborator Contribution | Pavel Shumyatsky is my research collaborator that I have visited a couple of times. He also be visited me in April 2019 for about 2 weeks. |

Impact | E. Khukhro, P. Shumyatsky and G. Traustason , Right Engel-Type subgroups and length parameters of finite groups, J. Aust. Math. Soc. 109 (2020), no. 3, 340-350. P. Longobardi, M. Maj, P. Shumyatsky, G. Traustason, Groups with boundedly many commutators of maximal order. J. Algebra 567 (2021), 269-283. |

Start Year | 2017 |

Description | Conference in Bath |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Other audiences |

Results and Impact | A conference on Engel Conditions in Groups held at the University of Bath in April 2019. This brought in research collaborators in my area and the proceedings will be published in the International Journal of Group Theory. |

Year(s) Of Engagement Activity | 2019 |