# On Cherlin's conjecture for finite binary primitive permutation groups

Lead Research Organisation:
University of South Wales

Department Name: Faculty of Computing, Eng. and Science

### Abstract

This research concerns the connection between the "local symmetry" and "global symmetry" of a mathematical object. To understand what this means let us consider the symmetries of a simple mathematical object: a regular hexagon.

If I randomly pick two different sides of this hexagon, then it is clear that they "look the same" -- this just follows from the fact that the hexagon is regular and so all sides have the same length. This is an example of a "local symmetry" -- that's the mathematical terminology for a situation where two portions of a mathematical object that look the same.

On the other hand, because the hexagon is regular there are many "global symmetries" -- these are transformations of my object so that after the transformation it still "looks the same". In the case of the regular hexagon, for instance, I can reflect the hexagon through a line connecting two opposite corners, and the resulting object will be the same as the one I started with. Likewise, I can rotate my hexagon by any multiple of 60 degrees and the same will be true -- these are all examples of global symmetries.

Now a natural question that mathematicians want to know when they study any given mathematical object is "when does the presence of a local symmetry imply the presence of a global symmetry?". For instance in the example above, it is clear that given any pair of edges on my hexagon, I can find a global symmetry (a rotation, for example) that moves the first edge to the second edge. Thus we could say that the local symmetry here is just a consequence of the global symmetry. In fact this is rather unusual: most mathematical objects will have many local symmetries that are NOT consequences of some global symmetry. A mathematical object for which all local symmetries derive from global symmetries is very special, and is called HOMOGENEOUS.

The research in this project concerns homogeneous RELATIONAL STRUCTURES. A relational structure is just a particular generalization of a network: take a bunch of "nodes" and connect them with "edges" and you have made yourself a network (think of cities connected by roads, or computers connected by wires for real-life examples). Indeed the hexagon can be thought of as a network -- each corner can be thought of as a node (there are 6 of these), and then there are 6 edges connecting the nodes. We would like to know which networks are homogeneous. To fully understand this question, one needs to be a bit careful about how we define the notion of "symmetry" for a network and there is not time to do this here. As a teaser, though, let us mention that the network given by a hexagon is NOT homogeneous, whereas the network given by a pentagon IS!

Finally, let us say a word about our methods: whenever one studies symmetry in mathematics, one is effectively doing GROUP THEORY. The set of symmetries of any mathematical object (say the reflections and rotations of our regular hexagon), is called the GROUP associated to the object. One can study this group "in the abstract", i.e. without really needing to study the object it is associated with. For instance, if I perform a particular reflection and then a particular rotation of my hexagon, I will end up with a new symmetry of the hexagon (in fact it will be another of the reflections) and I can think of this as a type of "multiplication" of my symmetries: I've "multiplied" two symmetries together and the result is a third symmetry. To fully describe the symmetry group of my hexagon I just need to write down the "multiplication table" of all pairs of symmetries.

A great deal is known about the structure of different groups. Indeed one of the most famous and important mathematical theorems is called THE CLASSIFICATION OF FINITE SIMPLE GROUPS and it describes the structure of an important class of groups. In this research we will use this theorem to study homogeneous relational structures; our aim is to classify an important subclass of these objects.

If I randomly pick two different sides of this hexagon, then it is clear that they "look the same" -- this just follows from the fact that the hexagon is regular and so all sides have the same length. This is an example of a "local symmetry" -- that's the mathematical terminology for a situation where two portions of a mathematical object that look the same.

On the other hand, because the hexagon is regular there are many "global symmetries" -- these are transformations of my object so that after the transformation it still "looks the same". In the case of the regular hexagon, for instance, I can reflect the hexagon through a line connecting two opposite corners, and the resulting object will be the same as the one I started with. Likewise, I can rotate my hexagon by any multiple of 60 degrees and the same will be true -- these are all examples of global symmetries.

Now a natural question that mathematicians want to know when they study any given mathematical object is "when does the presence of a local symmetry imply the presence of a global symmetry?". For instance in the example above, it is clear that given any pair of edges on my hexagon, I can find a global symmetry (a rotation, for example) that moves the first edge to the second edge. Thus we could say that the local symmetry here is just a consequence of the global symmetry. In fact this is rather unusual: most mathematical objects will have many local symmetries that are NOT consequences of some global symmetry. A mathematical object for which all local symmetries derive from global symmetries is very special, and is called HOMOGENEOUS.

The research in this project concerns homogeneous RELATIONAL STRUCTURES. A relational structure is just a particular generalization of a network: take a bunch of "nodes" and connect them with "edges" and you have made yourself a network (think of cities connected by roads, or computers connected by wires for real-life examples). Indeed the hexagon can be thought of as a network -- each corner can be thought of as a node (there are 6 of these), and then there are 6 edges connecting the nodes. We would like to know which networks are homogeneous. To fully understand this question, one needs to be a bit careful about how we define the notion of "symmetry" for a network and there is not time to do this here. As a teaser, though, let us mention that the network given by a hexagon is NOT homogeneous, whereas the network given by a pentagon IS!

Finally, let us say a word about our methods: whenever one studies symmetry in mathematics, one is effectively doing GROUP THEORY. The set of symmetries of any mathematical object (say the reflections and rotations of our regular hexagon), is called the GROUP associated to the object. One can study this group "in the abstract", i.e. without really needing to study the object it is associated with. For instance, if I perform a particular reflection and then a particular rotation of my hexagon, I will end up with a new symmetry of the hexagon (in fact it will be another of the reflections) and I can think of this as a type of "multiplication" of my symmetries: I've "multiplied" two symmetries together and the result is a third symmetry. To fully describe the symmetry group of my hexagon I just need to write down the "multiplication table" of all pairs of symmetries.

A great deal is known about the structure of different groups. Indeed one of the most famous and important mathematical theorems is called THE CLASSIFICATION OF FINITE SIMPLE GROUPS and it describes the structure of an important class of groups. In this research we will use this theorem to study homogeneous relational structures; our aim is to classify an important subclass of these objects.

### Planned Impact

The proposed research is in the area of pure mathematics and so, like most research in this area, it is likely that the main impact in the short term will be within the academic community. It is hard to judge what the longer term impacts might be but, in order to maximise the chances of impact to other areas, the results of the research will be publicised as widely as possible.

Within the academic community, the initial impact will be greatest amongst researchers working in areas connected to group theory and to model theory. As stated in the case, the proposed research encompasses, extends and organises existing results and conjectures that already exist in the literature, many of which have been the subject of intense study themselves, as well as being applied in work further afield.

Complete success in achieving the research objectives would have significant impact in the field of group theory. By unifying existing results it would round out a significant chapter in the area of permutation groups; equally, though, it would open the door towards exploiting model theoretic insights within group theory. It seems fair to say that, hitherto, the work of model theorists and group theorists has been held back somewhat by the technical difficulties inherent in converting model theoretic results on (say) structure theories into concrete theorems in group theory. Proving Cherlin's conjecture would demonstrate that such technical difficulties are not insurmountable and would, one hopes, open the way to further joint ventures between these two mathematical communities.

Beyond the immediate impact on the group theory community, one is tempted to speculate about the "cultural" impact of this sort of research on the mathematical research community. There have been, of course, a number of spectacular successes in the "export" of model theory to the wider mathematical community (perhaps most notably one has Hrushovski's proof of function-field Mordell-Lang, as well as Loeser and, later, Ngo's work on Langland's fundamental lemma). Still, these success notwithstanding, one might nonetheless speculate that the value of model theory has not been fully appreciated by all mathematicians: proving Cherlin's conjecture would give more evidence that the insights of model theory can have substantial impact on all areas of mathematics.

The proposed research also stands to have significant pedagogical impact: the proposed research has the advantage that it is perfectly possible to communicate many of the fundamental ideas to undergraduate students, and the area is a rich entry point for young mathematicians wishing to embark on a research career. The investigating team will make sure that the proposed research is communicated in as accessible a way as possible, and will take every opportunity to introduce new audiences to this research area.

The UK is currently world leading in the study of the interface between group theory and model theory, and success in the proposed research would further establish the UK's

reputation in this area and strengthen links with other research groups in Europe.

Within the academic community, the initial impact will be greatest amongst researchers working in areas connected to group theory and to model theory. As stated in the case, the proposed research encompasses, extends and organises existing results and conjectures that already exist in the literature, many of which have been the subject of intense study themselves, as well as being applied in work further afield.

Complete success in achieving the research objectives would have significant impact in the field of group theory. By unifying existing results it would round out a significant chapter in the area of permutation groups; equally, though, it would open the door towards exploiting model theoretic insights within group theory. It seems fair to say that, hitherto, the work of model theorists and group theorists has been held back somewhat by the technical difficulties inherent in converting model theoretic results on (say) structure theories into concrete theorems in group theory. Proving Cherlin's conjecture would demonstrate that such technical difficulties are not insurmountable and would, one hopes, open the way to further joint ventures between these two mathematical communities.

Beyond the immediate impact on the group theory community, one is tempted to speculate about the "cultural" impact of this sort of research on the mathematical research community. There have been, of course, a number of spectacular successes in the "export" of model theory to the wider mathematical community (perhaps most notably one has Hrushovski's proof of function-field Mordell-Lang, as well as Loeser and, later, Ngo's work on Langland's fundamental lemma). Still, these success notwithstanding, one might nonetheless speculate that the value of model theory has not been fully appreciated by all mathematicians: proving Cherlin's conjecture would give more evidence that the insights of model theory can have substantial impact on all areas of mathematics.

The proposed research also stands to have significant pedagogical impact: the proposed research has the advantage that it is perfectly possible to communicate many of the fundamental ideas to undergraduate students, and the area is a rich entry point for young mathematicians wishing to embark on a research career. The investigating team will make sure that the proposed research is communicated in as accessible a way as possible, and will take every opportunity to introduce new audiences to this research area.

The UK is currently world leading in the study of the interface between group theory and model theory, and success in the proposed research would further establish the UK's

reputation in this area and strengthen links with other research groups in Europe.

### Publications

Dalla Volta F
(2018)

*Cherlin's conjecture for sporadic simple groups*in Pacific Journal of Mathematics
GILL N
(2018)

*Cherlin's conjecture for almost simple groups of Lie rank 1*in Mathematical Proceedings of the Cambridge Philosophical Society
Gill Nick
(2016)

*Binary permutation groups: alternating and classical groups*in arXiv e-prints
N. Gill
(2021)

*Statistics for S_n acting on k-sets*Description | Cherlin's conjecture concerns the primitive actions of permutation groups -- thanks to work of Cherlin himself, and of Wiscons, it is enough for us to study almost simple groups. In the first year of this award, three relevant papers have been published -- all involving two of the investigators, Gill and Spiga; as well as Hunt (in one paper) and Dalla Volta (in another). See the portfolio for these. Thanks to these papers, in order to prove Cherlin's conjecture, two important families of groups need to be studied. Our job, for the remainder of this award, is to analyse these families and prove the conjecture in each case. The status of these families at the time of writing is as follows: 1. THE EXCEPTIONAL GROUPS OF LIE TYPE: (Feb 2021) We are close to completing a full draft of the proof in this case. There are a number of remaining open cases that we must deal with before we can claim a proof for this family of groups, but we have made substantial headway. It is our expectation that we will have a full proof in the next couple of months, albeit possibly allowing for a finite number of group actions to remain open. (Feb 2020) The exceptional groups case is now complete. A full proof has been written down, and requires only a final proof read. (Feb 2021) The proof has been checked and is complete. 2. THE CLASSICAL GROUPS OF LIE TYPE: (Feb 2019) In this case there are nine families of primitive action which we must analyse -- these correspond to Aschbacher's nine families of maximal subgroup. One of these families -- C1 -- has already been completed by two of the investigators, Gill & Spiga, and this work was published in the first year of the award. A second family -- C2 -- has now been completed, and we have a full draft proof in this case. We have also made substantial headway with respect to families C3, C5, C8 and C9, although we are some distance from a full proof in these cases. Once the exceptional groups are completed, these families will be our next priority. (Feb 2020) This case is on the cusp of completion: in principle all that remains are a number of computer calculations. We say "in principle" because this proof has not been thoroughly checked yet, and it is possible that some unexpected issues may arise. However, if all goes to plan, we expect this case to be fully complete within the next 2 months. The two results discussed above have been written up together in monograph form, together with an opening chapter aimed at communicating the maths behind Cherlin's Conjecture to group theorists, and other researchers in algebra. We expect this monograph to be ready for submission in the next 2 months. (Feb 2021) Covid-19 has delayed the completion of work on the classical groups. At this stage we have a short list of edits & computer calculations that still need to be made. We expect this task to be completed soon.... But I hesitate to give a date given that my previous projection in Feb 2020 proved to be inaccurate. COMPLEMENTARY WORK In the 8 months since the award has completed, Gill and Spiga have produced two complementary outputs in collaboration with their PhD student, Bianca Loda. Both of these concern primitive permutation groups with LARGE relational complexity. (For clarity: the topic of this grant is about establishing which primitive permutation groups have the SMALLEST possible relational complexity.) |

Exploitation Route | Our expectation is that a proof of Cherlin's conjecture will open new horizons at the juncture of group theory and model theory. Both the result itself, and the methods developed to prove the result, have substantial implications in both fields. Cherlin's beautiful paper "Sporadic homogeneous structures" gives a clear outline of some of these implications. |

Sectors | Other |

Description | Contributed talk to BIRS workshop, Canada |

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

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | This was an invited talk entitled "On the relational complexity of a finite permutation group". In this talk we announced a proof of Cherlin's conjecture for exceptional groups. |

Year(s) Of Engagement Activity | 2019 |

URL | https://www.birs.ca/events/2019/5-day-workshops/19w5046/videos/watch/201908261531-Gill.html |

Description | Seminar at University of Birmingham |

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

Part Of Official Scheme? | No |

Geographic Reach | Regional |

Primary Audience | Professional Practitioners |

Results and Impact | PI gave a talk in Dec 2018 entilted "On Cherlin's conjecture and sporadic structures". |

Year(s) Of Engagement Activity | 2018 |

Description | Seminar at University of Lincoln |

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

Part Of Official Scheme? | No |

Geographic Reach | Regional |

Primary Audience | Professional Practitioners |

Results and Impact | PI gave a talk in Nov 2018 entitled "On Cherlin's conjecture and sporadic structures". |

Year(s) Of Engagement Activity | 2018 |

Description | Seminar at the University of Southampton |

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

Part Of Official Scheme? | No |

Geographic Reach | Regional |

Primary Audience | Professional Practitioners |

Results and Impact | PI gave a talk in May 2019 entitled "On Cherlin's Conjecture". |

Year(s) Of Engagement Activity | 2019 |

Description | Seminar to the London Algebra Colloquium |

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

Part Of Official Scheme? | No |

Geographic Reach | National |

Primary Audience | Professional Practitioners |

Results and Impact | The PI gave a talk entitled "Some statistics for finite permutation groups" |

Year(s) Of Engagement Activity | 2019 |