# Group Generation: From Finite To Infinite

Lead Research Organisation:
University of St Andrews

Department Name: Mathematics and Statistics

### Abstract

Have you ever marvelled at the stunning symmetry of a butterfly? Or been frustrated by a photograph taken slightly off-centre? As humans we're attracted to symmetry and we encounter it every day in nature, art and architecture. It's no surprise, therefore, that symmetry plays a fundamental role across all sciences. For instance, it's the mathematical theory of symmetry that explains the tragic side-effects of the drug thalidomide seen in the 1960s. It's symmetry that provides the language for the Standard Model of theoretical physics. It's symmetry that's at the heart of novel cryptography that remains secure in an era of quantum computers.

Group theory is the area of mathematics dedicated to discovering general results that apply to all types of symmetry in all contexts, from the three-dimensional shapes of molecules, to concepts from physics in 196,883 dimensions, to abstract objects admitting no simple geometric interpretation. Indeed, one fruitful way to study an object is to consider the group of all its symmetries.

Just as Lego constructions can be broken into Lego bricks and as molecules can be broken into atoms, the group of symmetries of an object can be broken into smaller indivisible "simple groups". One of the greatest mathematical achievements of the twentieth century was the effort of hundreds of mathematicians across the world to classify all the finite simple groups.

For decades, mathematicians have been interested in when one can obtain all an object's symmetries by repeatedly combining two well-chosen symmetries. This is called "generation", and it has yielded surprising results, with links across mathematics. For example, Liebeck and Shalev proved that "almost all" pairs of symmetries in a finite simple group generate the entire group. Moreover, just last year, Burness, Guralnick and I gave a complete classification of the finite groups where every symmetry (other than the "do nothing" symmetry) can be matched with another with which it generates the entire group of symmetries.

However, these developments all concern groups of objects with a finite number of symmetries, but objects with infinitely many symmetries are very important in contemporary mathematics. My proposal is to begin a new programme of research to generalise developments on generation to the infinite.

More precisely, I seek to investigate whether the startling generation properties of the finite simple groups hold for the infinite simple groups such as Thompson groups and related groups of homeomorphisms of Cantor space, with a view to forming a deeper understanding of the generation properties of finitely presented infinite simple groups. In addition, by exploiting recent developments in the theory of finite (almost) simple groups, I will address open questions regarding the generation of finite groups.

I propose carrying out this research at the University St Andrews, which is home to a number of leading researchers in both finite and infinite groups. Moreover, it hosts CIRCA, a research centre joint between mathematics and computer science. This highlights potential applications of the proposed programme of work: from cryptographers to chemists, researchers carry out computer calculations involving symmetry, and knowing that all the symmetries of an object can be generated by just two provides an efficient way to carry out many of these computations.

Group theory is the area of mathematics dedicated to discovering general results that apply to all types of symmetry in all contexts, from the three-dimensional shapes of molecules, to concepts from physics in 196,883 dimensions, to abstract objects admitting no simple geometric interpretation. Indeed, one fruitful way to study an object is to consider the group of all its symmetries.

Just as Lego constructions can be broken into Lego bricks and as molecules can be broken into atoms, the group of symmetries of an object can be broken into smaller indivisible "simple groups". One of the greatest mathematical achievements of the twentieth century was the effort of hundreds of mathematicians across the world to classify all the finite simple groups.

For decades, mathematicians have been interested in when one can obtain all an object's symmetries by repeatedly combining two well-chosen symmetries. This is called "generation", and it has yielded surprising results, with links across mathematics. For example, Liebeck and Shalev proved that "almost all" pairs of symmetries in a finite simple group generate the entire group. Moreover, just last year, Burness, Guralnick and I gave a complete classification of the finite groups where every symmetry (other than the "do nothing" symmetry) can be matched with another with which it generates the entire group of symmetries.

However, these developments all concern groups of objects with a finite number of symmetries, but objects with infinitely many symmetries are very important in contemporary mathematics. My proposal is to begin a new programme of research to generalise developments on generation to the infinite.

More precisely, I seek to investigate whether the startling generation properties of the finite simple groups hold for the infinite simple groups such as Thompson groups and related groups of homeomorphisms of Cantor space, with a view to forming a deeper understanding of the generation properties of finitely presented infinite simple groups. In addition, by exploiting recent developments in the theory of finite (almost) simple groups, I will address open questions regarding the generation of finite groups.

I propose carrying out this research at the University St Andrews, which is home to a number of leading researchers in both finite and infinite groups. Moreover, it hosts CIRCA, a research centre joint between mathematics and computer science. This highlights potential applications of the proposed programme of work: from cryptographers to chemists, researchers carry out computer calculations involving symmetry, and knowing that all the symmetries of an object can be generated by just two provides an efficient way to carry out many of these computations.

### Publications

Harper S
(2023)

*The maximal size of a minimal generating set*in Forum of Mathematics, SigmaDescription | Derangements in finite groups |

Organisation | University of Bristol |

Department | School of Mathematics |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | Research contribution to a manuscript in progress on this subject. |

Collaborator Contribution | Research contribution to a manuscript in progress on this subject. |

Impact | Manuscript in progress. |

Start Year | 2021 |