- W/Prof Cheryl Praeger AM FAA
- Prof. Dr. Gerhard Hiss

Jun 2010

Sep 2015

2-generated irreducible subgroups of quasi-simple groups

Our focus is on matrix groups, that is subgroups of the finite general linear group GL(d,q), the group of non-singular (d x d)-matrices over a finite field of order q. In such groups, we are investigating elements which leave invariant, and act irreducibly on, large dimensional subspaces.

Over the past years one of the main focus points of computational group theory has been the challenging task to design algorithms for analysing the structure of matrix groups effectively.

Matrix groups, even in relatively small dimensions, can have a very large number of elements. Therefore, it is advantageous to create algorithms which are randomised in the sense that they derive information about a matrix group by looking at properties of independent, uniformly distributed random elements.

One property that an element might have and which has had important algorithmic applications, is whether the element order is divisible by certain primes called primitive prime divisors. Such primitive prime divisor elements, or ppd-elements for short, have been exploited for example by Niemeyer and Praeger in 1997 to recognise finite classical groups in their natural representation.

The ppd-elements used in the Niemeyer-Praeger algorithm have the following property. They leave invariant and act irreducibly on large dimensional subspaces. Our research develops theoretical results which open up the possibility to waive the restriction of looking for ppd-elements and to evolve a new generation of (even more efficient) algorithms based solely on elements with large irreducible subspaces.