5-day minicourse on   New Approaches to Computing Finite Group Invariants by Prof. John J. Cannon, University of Sydney, Australia

Prof. John Cannon  was awarded a PhD by the University of Sydney in 1970 for a thesis on Computational Algebra. He currently holds a Personal Chair in Mathematics at Sydney. Much of his research is concerned with the design of efficient algorithms for determining structural properties of finitely generated algebraic structures. In joint work over the past ten years with Derek Holt he has developed new algorithms for the fundamental invariants of a finite group that makes possible the analysis of groups of a complexity far beyond what was possible previously. Cannon has also undertaken the development of two widely-used Computer Algebra systems, CAYLEY and MAGMA, designed for computations at a high level of abstraction in algebra, number theory and geometry. Cannon has been an Associate Editor of the Journal of Symbolic Computation since its foundation and is on the editorial boards of Applicable Algebra and the Australasian Journal of Combinatorics. He is Chair of the organising committee for the Fifth Algorithmic Number Theory Conference to be held in Sydney, July 2002. He is the recipient of two major Australian scientific awards for his contributions to Computer Algebra: the CSIRO medal in 1993 and the Clunies-Ross National Science and Technology Award in 2001.

Contents
The Classification Theorem for Finite Simple Groups allows us, in principle, to reduce many questions about an arbitrary finite group to the consideration of particular known simple groups (its simple constituents).  The design of efficient algorithms based on the idea of "reduction to a simple group" represents a considerable challenge to the algorithm designer and it is only very recently that efficient practical algorithms have started to emerge. These lectures will describe the strategy behind a family of such reduction algorithms devised by Derek Holt and Cannon for computing basic properties of a finite group G (e.g., normal subgroups, subgroup lattice, automorphism group). There are three major steps in such a reduction algorithm:

• Determining the composition factors of G.
• Solving the problem for each of the simple constituents of G.
• Lifting the answer back to G.

The course will begin with a discussion of the basic algorithms for permutation groups. This will be followed by a review of computational techniques in the representation theory of groups. Algorithms for computing composition factors, composition series and chief series based on the O'Nan-Scott Theorem will be presented next. As step c) is heavily dependent upon what is to be computed, we will examine its solution in the case of conjugacy classes of elements, normal subgroups and conjugacy classes of subgroups. An overview of similar algorithms for problems such testing groups for isomorphism will be given.

In addition to the theory lectures, some applications of these techniques will be presented by various guest lecturers. These are expected to include:

• The construction of incidence geometries for finite simple groups [Dr D Leemans].
• The computation of the Galois group of a polynomial.
• The classification of the primitive groups having degree up to 256.

Assumed knowledge
For each of the three topics below there are many suitable books. The books listed are meant to be indicative of necessary background
knowledge:

• General Group Theory: J. Rotman : The Theory of Groups: An Introduction", Allyn and Bacon, Boston, 1978. [Chapters 1--8].
• Permutation Groups:  J. Dixon and B Mortimer, "Permutation Groups", Springer, 1996. [Chapters 1--4].
• Representation Theory: G. James and M. Liebeck, "Representations and Characters of Groups", Cambridge, 1993. [ Chapters 1--13].

• Group Theory Algorithms: G. Butler, "Fundamental Algorithms for Permutation Groups", LNCS 559, Springer, 1991.

Admission fee
The admission fee for a minicourse is Euro 680 (NLG 1500). However, our courses are free of charge for EIDMA members and for university students. Reductions or exemptions do apply to members of other academic institutes.

Registration
You can register by sending an e-mail to Dimitri Leemans at the ULB: dleemans@ulb.ac.be
 

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