Engel Type Conditions in Groups
Lead Research Organisation:
University of Bath
Department Name: Mathematical Sciences
Abstract
The project falls within the area of Group Theory that is a sub branch of Algebra. The supervisor currently holds a 3
year EPSRC grant. The project is on `Left 3-Engel elements in groups' and is centred around the challenging open
question, whether a left 3-Engel element is always contained in the locally nilpotent radical. There are various
questions linked to this project that would provide a starting point for my research in this area. The topic is of central
importance within group theory and successful research in this area would thus have impact on the field.
Training during the first months will involve much background reading on topics like `Lie algebras', `Lie ring methods
in group theory', `Engel groups' and `p-groups'.
The aim of the project will be to contribute to a better understanding of Engel conditions in groups that might include
left 3-Engel elements. The methodology includes reading up what is currently known, that includes some important
recent developments, as well as understanding the various techniques that have been applied successfully in the
past.
year EPSRC grant. The project is on `Left 3-Engel elements in groups' and is centred around the challenging open
question, whether a left 3-Engel element is always contained in the locally nilpotent radical. There are various
questions linked to this project that would provide a starting point for my research in this area. The topic is of central
importance within group theory and successful research in this area would thus have impact on the field.
Training during the first months will involve much background reading on topics like `Lie algebras', `Lie ring methods
in group theory', `Engel groups' and `p-groups'.
The aim of the project will be to contribute to a better understanding of Engel conditions in groups that might include
left 3-Engel elements. The methodology includes reading up what is currently known, that includes some important
recent developments, as well as understanding the various techniques that have been applied successfully in the
past.
| Description | My research is on Engel groups and Engel elements. Engel Theory and Engel groups are closely related to the well-known Burnside Problems. The Burnside problem was posed in 1902 and has remained one of the most important questions in Group Theory. There is a strong connection between Burnside problems and Engel problems which makes Engel theory very attractive for algebraists to study. The Burnside Problem asks if every finitely generated group, in which x^n=1, for all x, must be finite (true for n=2,3,4,6, not for large n). For example, if one looks at the group G generated by two elements, in which all elements x satisfy x^8=1, then it is still unknown whether this will be finite or not. The Engel analogue asks whether a finitely generated group in which [[[[y,x],x]....],x]=1, where we commute with x by n times, for all x,y in G (weak local commutativity property) must be nilpotent, i.e. [x_1, ..., x_m]=1 for some m (weak global commutativity property). This is known to be true for values of n<5, however it still remains unsolved in general and makes this one of the most important questions in Engel theory. My PhD research was concerning left 3-Engel elements in locally finite p-groups with the central question: Is every left 3-Engel element always in the locally nilpotent radical of the group? In other words, is it true that the normal closure of a left 3-Engel element is locally nilpotent? It is know that this holds for left 2-Engel elements shown by Burnside. More precisely, Burnside showed that the normal closure of a left 2-Engel element is abelian and therefore it lies in the locally nilpotent radical of the group. It is also known that for very large values of n this is not true, thus the question now is what is the smallest n for which a left n-Engel element is not in the locally nilpotent radical, hence the central question of our research. Major advances have been made towards answering that question. Trausason and Jabara have shown that left 3-Engel elements of odd order are contained in the locally nilpotent radical. Moreover Traustason and Tracey showed that in order to generalise this to left 3-Engel elements of any finite order it suffices to only deal with elements of order 2. In my PhD we focused on a result by Martin Newell where he showed that right 3-Engel elements are always in the locally nilpotent radical. In particular he showed that the normal closure of a right 3-Engel element is nilpotent of class at most 3. It was only reasonable to ask if the analogue holds for left 3-Engel elements. We proved that this is not the case by finding an infinite family of locally finite 2-groups G with a left 3-Engel element x such that the normal closure of x in G is not nilpotent. Moreover we showed that when dealing with p-groups where p is an odd prime, we can give an example of a locally finite p-group G containing a left 3-Engel element x, where the normal closure of x in G is not nilpotent. We are currently looking at groups with left 2-Engel elements of order 2 and looking for an example where the normal closure is not locally nilpotent. This is a joint work with Sandro Mattarei and Gunnar Traustason. |
| Exploitation Route | The question whether left 3-Engel elements are in the locally nilpotent radical of the group is a longstanding one and answering it would be a major breakthrough in this area. Also, going back to the example of a finitely generated group G, where all elements x satisfy x^8=1, it is still unknown whether this is finite or not. If, however we assume that all left 4-Engel elements in G are in the locally nilpotent radical of G, then it will follow that indeed groups of exponent 8 are locally finite, which would give the answer to a part of the Burnside problem. So understanding what happens for left 3-Engel elements would be an important step in that direction. |
| Sectors | Other |
| Description | Left 3-Engel elements of order 2 |
| Organisation | University of Lincoln |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | This is ongoing joint project towards answering whether there exists an example of a left 3-Engel element not in the locally nilpotent radical of the group. |
| Collaborator Contribution | This is ongoing joint project towards answering whether there exists an example of a left 3-Engel element not in the locally nilpotent radical of the group. |
| Impact | Still in its early stages. |
| Start Year | 2022 |
| Description | Left 3-Engel elements on locally finite p-groups |
| Organisation | University of Salerno |
| Country | Italy |
| Sector | Academic/University |
| PI Contribution | Under this partnership we showed that for any odd prime p there exists a locally finite p-group containing a left 3-Engel element whose normal closure is not nilpotent. |
| Collaborator Contribution | Under this partnership we showed that for any odd prime p there exists a locally finite p-group containing a left 3-Engel element whose normal closure is not nilpotent. |
| Impact | We published the paper (arXiv:2007.09882) published in International Journal of Algebra and Computation, IJAC's Volume No. 31, Issue No. 01, pp. 2020 - 160, Year 2020. |
| Start Year | 2019 |