Derek F. Holt

Testing modules for irreducibility

A practical method is described for deciding whether or not a finite-dimensional module for a group over a finite field is reducible or not. In the reducible case, an explicit submodule is found. The method is a generalistaion of the Parker-Norton ‘Meataxe’ algorithm, but it does not depend for its efficiency on the field being small. The principal tools involved are the calculation of the nullspace and the characteristic polynomial of a matrix over a finite field, and the factorisation of the latter. Related algorithms to determine absolute irreducibility and module isomorphism for irreducibles are also described. Details of an implementation in the GAP system, together with some performance analyses are included.

The use of Knuth-Bendix methods to solve the wordproblem in automatic groups

Certain classes of infinite groups arising from geometry and topology are known to have solvable word problem. We describe the development of practical methods for the solution of the word problem based on the reduction of words in the generators to a normal form. The Knuth-Bendix completion procedure is the principal tool used but, in the case that this process does not halt, we use alternative methods involving the construction of finite-state automata. A computer implementation of these procedures together with some performance statistics on some simple examples are also described.

Groups with Context-Free Co-Word Problem

The class of co-context-free groups is studied. A co-context-free group is defined as one whose co-word problem (the complement of its word problem) is context-free. This class is larger than the subclass of context-free groups, being closed under the taking of finite direct products, restricted standard wreath products with context-free top groups, and passing to finitely generated subgroups and finite index overgroups. No other examples of co-context-free groups are known. It is proved that the only examples amongst polycyclic groups or the Baumslag-Solitar groups are virtually abelian. This is done by proving that languages with certain purely arithmetical properties cannot be context-free; this result may be of independent interest.

Automorphism group computation and isomorphism testing in finite groups

A new method for computing the automorphism group of a finite permutation group and for testing two such groups for isomorphism is described. Some performance statistics are included for an implementation of these algorithms in the Magma language.

Recognizing badly presented Z-modules

Finitely generated Z-modules have canonical decompositions. When such modules are given in a finitely presented form, there is a classical algorithm for computing a canonical decomposition. This is the algorithm for computing the Smith normal form of an integer matrix. We discuss algorithms for Smith-normal-form computation, and present practical algorithms which give excellent performance for modules arising from badly presented abelian groups. We investigate such issues as congruential techniques, sparsity considerations, pivoting strategies for Gauss-Jordan elimination, lattice basis reduction, and computational complexity. Our results, which are primarily empirical, show dramatically improved performance on previous methods.

Computing maximal subgroups of finite groups

Abstract 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.

Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic

The classification of the nilpotent orbits in the Lie algebra of a reductive algebraic group (over an algebraically closed field) is given in all the cases where it was not previously known (E 7 and E 8 in bad characteristic, F 4 in characteristic 3). The paper exploits the tight relation with the corresponding situation over a finite field. A computer is used to study this case for suitable choices of the finite field.

Computing the Subgroups of a Permutation Group

A new method for computing the conjugacy classes of subgroups of a finite group is described.

Artin groups of large type are shortlex automatic with regular geodesics

We prove that any Artin group of large type is shortlex automatic with respect to its standard generating set, and that the set of all geodesic words over the same generating set satisfies the Falsification by Fellow-Traveller Property (FFTP) and hence is regular.

THE LINEARITY OF THE CONJUGACY PROBLEM IN WORD-HYPERBOLIC GROUPS

The main result proved in this paper is that the conjugacy problem in word-hyperbolic groups is solvable in linear time. This is using a standard RAM model of computation, in which basic arithmetical operations on integers are assumed to take place in constant time. The constants involved in the linear time solution are all computable explicitly. We also give a proof of the result of Mike Shapiro that in a word-hyperbolic group a word in the generators can be transformed into short-lex normal form in linear time. This is used in the proof of our main theorem, but is a significant theoretical result of independent interest, which deserves to be in the literature. Previously the best known result was a quadratic estimate.