Probabilistic and homological methods in group theory

Lead Research Organisation: University of Birmingham
Department Name: School of Mathematics

Abstract

The project proposes to investigate a series of question related to generation properties of finite simple groups using both homological properties and probabilistic methods. In particular we will consider questions realted to small presentations will be investigated using estimates on chomology of the group and questions about identification of the subgroup structure via cocliques in the generating graph.

Publications

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Saunders, J P (2019) Maximal cocliques in PSL(2,q) in Communications in Algebra

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509590/1 01/10/2016 30/09/2021
1809338 Studentship EP/N509590/1 01/10/2016 30/03/2020 Jack Philip Saunders
 
Description The initial research undertaken on generation of finite simple groups resulted in a paper which appeared in Communications in Algebra in 2019. We classified the largest examples of a particular kind of subset of a particular class of groups in such a way as to highlight a link between the structure of these groups and the structure of a certain graph corresponding to these groups.

The second part of the research, focusing on the dimension of the cohomology of finite simple groups (roughly, how one can 'stick' other groups to them in a particular way) has resulted in a complete understanding of this cohomology in two of the 'smallest' families of finite simple groups and an approach which generalises to all groups in a certain sense. We have a very general result relating higher-dimensional cohomology to the cohomology of some module in a dimension one smaller.

We have also written certain functions in the computer algebra system MAGMA in order to construct certain representations of Chevalley groups.
Exploitation Route The work on generation of finite simple groups may reduce the task of finding 'cocliques' (or independent sets) in the aforementioned graphs to finding maximal subgroups of these groups, or vice versa dependent on which problem happens to be easiest. Further investigation into these methods, for example with other classes of groups, may result in further links.

The work on cohomology generalises to wider classes of groups. The general result mentioned earlier may help other people research in the same area if they are able to find out more information about the structure to which we have related the higher-dimensional cohomology.

The MAGMA code, while not yet implemented anywhere, may be of use to the MAGMA team themselves and I have been contacted by one of the developers to discuss potential implementation of these functions or something similar in MAGMA.
Sectors Other

URL https://arxiv.org/abs/2002.04183