# Bounding the degree of permuation representations of quotient groups of finite permuation groups

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

This project is in the EPSRC Research Area of Algebra and, more specifically, in Group Theory, and closely related to and motivated by Computational Group Theory.

Over the past 50 years, many efficient algorithms have been devised, analysed and implemented for computing with finite permutation groups (i.e. subgroups of finite symmetric groups Sym(n)). Since quotient groups form a fundamental component of group theory, it is natural to ask, for a subgroup G of Sym(n), what is the smallest m such that a quotient group G/N of G arises as a subgroup of Sym(m). Unfortunately, m can be exponentially larger then n in general. On the other hand, many specific instances have been identified in which m is at most n - this occurs for example when N is the largest normal solvable subgroup of G.

In this project, the student will attempt to find further conditions under which m is at most n or, less ambitiously, when m can be bounded by a polynomial function of n. One such condition that is worth investigating is when the composition factors of N are all nonabelian.

There are also some interesting examples in the other direction, where N is very small, G/N has a small degree permutation representation, but G does not. An example of this is when G = Alt(n) or Sym(n) and |N| = 2 with N in [G,G]. It will be still be interesting to determine the smallest degree permutation representation of G in such cases, if only to find out just how bad things can get.

Any new results along these lines are likely to result in improvements to algorithms for computing in finite groups which, in turn, via widely used software packages, such as Sage, GAP, and Magma, have applications to all aspects of computing in discrete mathematics.

Over the past 50 years, many efficient algorithms have been devised, analysed and implemented for computing with finite permutation groups (i.e. subgroups of finite symmetric groups Sym(n)). Since quotient groups form a fundamental component of group theory, it is natural to ask, for a subgroup G of Sym(n), what is the smallest m such that a quotient group G/N of G arises as a subgroup of Sym(m). Unfortunately, m can be exponentially larger then n in general. On the other hand, many specific instances have been identified in which m is at most n - this occurs for example when N is the largest normal solvable subgroup of G.

In this project, the student will attempt to find further conditions under which m is at most n or, less ambitiously, when m can be bounded by a polynomial function of n. One such condition that is worth investigating is when the composition factors of N are all nonabelian.

There are also some interesting examples in the other direction, where N is very small, G/N has a small degree permutation representation, but G does not. An example of this is when G = Alt(n) or Sym(n) and |N| = 2 with N in [G,G]. It will be still be interesting to determine the smallest degree permutation representation of G in such cases, if only to find out just how bad things can get.

Any new results along these lines are likely to result in improvements to algorithms for computing in finite groups which, in turn, via widely used software packages, such as Sage, GAP, and Magma, have applications to all aspects of computing in discrete mathematics.

## People |
## ORCID iD |

Derek Holt (Primary Supervisor) | |

Robert Chamberlain (Student) |

### Publications

CHAMBERLAIN R
(2018)

*MINIMAL EXCEPTIONAL -GROUPS*in Bulletin of the Australian Mathematical Society### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/N509796/1 | 01/10/2016 | 30/09/2021 | |||

1789299 | Studentship | EP/N509796/1 | 03/10/2016 | 31/03/2020 | Robert Chamberlain |

Description | Given a finite group G the object of study in my research is the smallest n such that G can be found as a subgroup of the symmetric group Sym(n). One of the main objectives is given a quotient G/N of G to find new results on the smallest m for which G/N can be found as a subgroup of Sym(m). If m is greater than n we call G exceptional. To this end I have two major results: (published) Given a prime p the smallest exceptional group of order p^m has order p^5. This generalises the case p=2 which was already shown by Easdown and Praeger. (published) If N has no abelian composition factors then m is at most n. This area of study mentioned in the Award Abstract. We have also calculated the least n such that G can be found as a subgroup of Sym(n) where G/N=Alt(d), N is contained in [G,G] and |N|=2. This is also mentioned in the Award Abstract. We have also calculated the least n such that G can be found as a subgroup of Sym(n) where G=SL(d,q) or G is the Schur cover of a sporadic simple group (except for a few cases beyond current computational power). |

Exploitation Route | Finite groups are often represented on computers as subgroups of Sym(n) for some n (also known as permutation groups). The smaller n is the faster algorithms which work with permutation groups run. Any results concerning the smallest such n may therefore result in faster group theoretic computations and can be implemented in existing software packages such as Sage, GAP and Magma. Developing a better understanding of the least n such that G/N arrises as a subgroup of Sym(n) when G is a group of prime power order will help answer a number of open research questions. For example, it is conjectured that if G is a subgroup of Sym(n) and G/N is abelian then G/N arrises as a subgroup of Sym(n). It is known that, among other properties, a smallest counterexample to this must be some G of prime power order. Following my result on G/N when N has no abelian composition factors, bounding the least n such that G/N arrises as a subgroup of Sym(n) in general may be reduced to the case N is elementary abelian (a direct product of abelian groups). As such my research will likely appear in a future work finding such n. Some of my results fit into a larger research problem - computing the least n such that G can be found as a subgroup of Sym(n) where G is a quasisimple group. The only remaining cases are classical groups and a few Schur covers of large sporadic simple groups. It may be possible to use a similar approach to the one I have used to work through these systematically. |

Sectors | Other |