J. Neubüser

On integral groups

SummaryAn algorithm for the determination of all integral classes of reducible integral matrix groups of given dimension from those of lower dimension is described. For dimensionn=4 there are 567 such classes.

Some remarks on the computation of conjugacy classes of soluble groups

Laue et al have described basic algorithms for computing in a finite soluble group G given by an AG-presentation, among them a general algorithm for the computation of the orbits of such a group acting on some set Ω. Among other applications, this algorithm yields straightforwardly a method for the computation of the conjugacy classes of elements in such a group, which has been implemented in 1986 in FORTRAN within SOGOS by the first author and in 1987 in C within CAYLEY. However, for this particular problem one can do better, as discussed in this note.

On crystallography in higher dimensions. I. General definitions

For use in subsequent parts of this series, some main concepts of mathematical crystallography (arithmetic crystal class, geometric crystal class, lattice, Bravais type, crystal family, holohedry, crystal system) are defined algebraically.

Investigations of groups on computers

Publisher Summary This chapter presents a survey of methods used in and results obtained by programs for the investigation of groups. It discusses the construction of all groups of a particular kind. A program of this kind was first suggested as early as 1951 by M. H. A. Newman for the investigation of the groups of order 256. P. Hall introduced the concept of isoclinism for the classification and construction of p-groups. Newman pointed out that the number of cases to be investigated for the determination of all groups of order 256 in a simple-minded use of Halls ideas would be far too big for computers then. He gave an estimate to show that by a probabilistic approach it would be feasible to obtain the great majority of these groups in a reasonable time. It seems, however, that this suggestion has never been followed. For the search for simple groups, so far systematic searches with computers have established only the nonexistence of simple groups of certain kinds.

On integral groups. III. Normalizers

Methods for determining a generating set for the normalizer of a finite group of n X n integral matrices, i.e., an n-dimensional crystallographic point group, are dis- cussed. Necessary and sufficient conditions for the finiteness of such a normalizer are derived, and several examples of the application of the methods to cases when the normalizer is infinite are presented.

On Finite Groups With ‘Hidden’ Primes

The starting point of this investigation was a question put to us by Martin B. Powell: If the prime number p divides the order of the finite group G , must there be a minimal set of generators of G that contains an element whose order is divisible by p ? A set of generators of G is minimal if no set with fewer elements generates G . A minimal set of generators is clearly irredundant, in the sense that no proper subset of it generates G ; an irredundant set of generators, however, need not be minimal, as is easily seen from the example of a cyclic group of composite (or infinite) order. Powells question can be asked for irredundant instead of minimal sets of generators; it turns out that the answer is not the same in these two cases. A different formulation, together with some notation, may make the situation clearer.

On crystallography in higher dimensions. II. Procedure of computation in R4

The mathematical background and the computing methods applied to the classification of lattices and crystallographic groups of 4-dimensional space R4 are described.

On crystallography in higher dimensions. III. Results in R4

An explicit classification of lattices and crystallographic groups of 4-dimensional space R4 is given. There are (in R4): 710 arithmetic crystal classes; 227 geometric crystal classes belonging to 118 isomorphism types of groups; 64 Bravais classes corresponding to 64 Bravais types of lattices; 33 crystal systems; 23 crystal families.

Some applications of group theoretical programs

Both special purpose programs and more general systems have been implemented for the study of groups. The special programs were in most cases designed for calculations occurring in the investigation of very big finite groups, such as some of the newly discovered simple groups. The general systems in most cases allow a fairly detailed analysis of comparatively small groups only. Of course, general systems can be used to construct examples for teaching and for testing hypotheses, but often the more detailed structure of smaller subgroups of big groups, e.g., their Sylow subgroups, is needed. Another application of such a system, namely, to the classification of crystallographic groups and lattices in 4-dimensional space will be emphasized.

On crystallography in higher dimensions. I. General definitions. Corrigendum

It is pointed out that the definition of crystal system as given in Neubuser, Wondratschek & Billow [Acta Cryst. (1971), A27, 517-520] is not dimension-independent. Nevertheless it leads to no ambiguity for dimensions 1, 2, 3, and 4, which are the only ones in which it has been used in subsequent papers. An emendation will be given in Neubuser, Plesken & Wondratschek [match (Informal Commun. Math. Chem.) (1980), to be published].