John J. Cannon’s research while affiliated with The University of Sydney and other places

A complete procedure is described for constructing the irreducible KG-modules and their Brauer characters, where K is a finite field of characteristic p and G is a finite permutation or matrix group. The central idea is to construct a sequence {S1,…,Sn} of KG-modules, each having relatively small dimension, such that each Si has one or more irreducible constituents that are not constituents of S1,…,Si−1. The Meataxe, used in conjunction with condensation, is used to extract the new irreducibles from each Si. The algorithm has been implemented in Magma and is capable of constructing irreducibles of dimension over 200000.

We determine the minimal degree permutation representations of all finite groups with trivial soluble radical, and describe applications to structural computations in large finite matrix groups that use the output of the CompositionTree algorithm. We also describe how this output can be used to help find an effective base and strong generating set for such groups. We have implemented the resulting algorithms in Magma, and we report on their performance.

Let G be a finite non-solvable group. We prove that there exists a proper subgroup A of G such that G is the product of three conjugates of A, thus replacing an earlier upper bound of 36 with the smallest possible value. The proof relies on an equivalent formulation in terms of double cosets, and uses the following theorem which is of independent interest and wider scope: Any group G with a BN-pair and a finite Weyl group W satisfies $G=\left( Bn_{0}B\right) ^{2}=BB^{n_{0}}B$ where $n_{0}$ is any preimage of the longest element of W. The proof of the last theorem is formulated in the dioid consisting of all unions of double cosets of B in G. Other results on minimal length product covers of a group by conjugates of a proper subgroup are given.

We describe the theory and implementation in Magma of algorithms to compute the projective indecomposable KG-modules for finite groups G and finite fields K. We describe also how they may be used together with dimension-shifting techniques to compute cohomology groups Hn(G, M) for finite-dimensional KG-modules M and n3.

We describe our successful computation of a list of representatives of the 2,801,324 conjugacy classes of transitive groups of degree 32.

We classify the radical p-subgroups and chains of the Fischer simple group Fi 24 ' and then verify the Alperin weight conjecture and the Uno reductive conjecture for Fi 24 ' .

We describe a significantly improved algorithm for computing the conjugacy classes of a finite permutation group with trivial soluble radical. We rely on existing methods for groups that are almost simple, and we are concerned here only with the reduction of the general case to the almost simple case.

We describe a practical algorithm for computing representatives of the conjugacy classes of subgroups up to a given index in a nite group. This al- gorithm has been implemented in Magma, and we present some performance statistics.

Polycyclic generating sequences are known to be a powerful tool in the design of practical and efficient algorithms for computing in finite soluble groups. Here we describe a further development: the so-called special polycyclic generating sequences. We give an overview of their properties and introduce a practical algorithm for determining a special polycyclic generating sequence in a given finite soluble group.

We describe a practical algorithm for computing representatives of the conjugacy classes of maximal subgroups in a finite group, together with details of its implementation for permutation groups in the MAGMA system. We also describe methods for computing complements of normal subgroups and minimal supplements of normal soluble subgroups of finite groups.