Fusion Systems

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

Abstract

This pure mathematics project in algebra, studies Fusion systems on $p$-groups of odd order. These are still mysterious objects. The project will investigate Puig Solvable fusion systems with the intention of completing the classification started by Aschbacher.

Aschbacher's work solves the problem when $p=2$. The methods used will be fusion and amalgam theoretic. Once this part of the project is complete we will move on to study problems in fusion systems which have direct application to the problem of revising the classification of the finite simple groups.

Publications

10 25 50

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509590/1 01/10/2016 30/09/2021
1955508 Studentship EP/N509590/1 25/09/2017 31/03/2021 Martin Van Beek
 
Description We have generalized a result by Michael Aschbacher in 2010 ("Generation of fusion systems of characteristic 2-type") which only held for the prime number 2, to an appropriate analogue for all prime numbers. We are working towards (and believe it to be possible) to generalize this further in an attempt to classify a larger number of fusion systems. Indeed, the so-called "characteristic p-type" fusion systems form one of only two possible classes of fusion systems (the other being the so-called "component type fusion systems" which require different methods.) Our further generalization would expand the class of "characteristic p-type" while still primarily using the methods we have developed.

In addition to this, we have developed concrete methods to deal with certain kinds of fusion systems and, as a result, have a full classification of some historically stubborn cases (e.g. some of the rank 2 Lie type groups' associated fusion systems).
Exploitation Route We feel this should all contribute toward a revised proof of the "Classification of Finite Simple Groups." The revision project is a large undertaking and there is plenty of scope for future research problems in this area which should use the results we have developed so far.

In addition, some of our results and techniques may find use in the areas of "representation theory" and "algebraic topology" where fusion systems have also been utilized in recent years.
Sectors Other