COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

X Preface
I am indebted to the authors for their co-operation in producing this
volume, to Dr. Howlett for contributing the Foreword, to the S.R.C.
Atlas Computer Laboratory for a Research Fellowship during the tenure
of which much of the work of editing was done, and to the University of
Glasgow for leave of absence both to attend the Conference and to accept
the Research Fellowship. It is also a pleasure to acknowledge the co-operation
of the publishers, Pergamon Press, and it is through no fault of theirs
that events such as the devaluation of British currency (which necessitated
a change of printers) have conspired to delay the appearance of this volume.
JOHN LEECH
Investigations of groups on computers
J . NEUB~~SER
1. Introduction. In this paper a survey is given of methods used in and
results obtained by programmes for the investigation of groups. Although
the bibliographies [De l] and [Sa 1,2,3] have been used, among other
sources, no claim for completeness can be made for two reasons, that
some publications may have been overlooked, and that the conference
itself has shown once more that there are many activities in this field which
are not (yet) covered by regular publications.
1.1. Papers and programmes have not been included if their main objective
is something different from the study of groups, even if groups play
some role in them. Four particular cases of this kind may be mentioned.
1.1.1. Combinatorial problems dealing with things like generation of
permutations, graphs, orthogonal latin squares, projective planes, block
designs, difference sets, and Hadamard matrices. For most of these topics
surveys are available, e.g. [Ha 1,4; SW 11.
13.2. Theorem-proving programmes. Most of these have been used to
construct proofs for very elementary group-theoretical theorems. There
seems to be only one mo l] specifically made to handle group-theoretical
statements.
1.1.3. Programmes for the determination and study of homology and
homotopy groups, where the main interest is in the topological relevance
of the results. Such papers are [Li 4; Ma 1, 2; Pi l] and part of [Ca 23.
1.1.4. Applications of groups in fields like coding theory [Pe l] or the
use of a computation in residue class groups for the improvement of a
programme described in pa 11.
1.2. Although the distinction is not always quite clear cut, it is practical
for this survey to distinguish between special purpose and general purpose
programmes. In spite of the fact that the first category is more likely to
produce significant contributions to group theory, more space will be
given in this report to the second kind, simply because this is the author’s
own field of work.
2. Special purpose programmes. By the first kind I mean programmes
specially made for the investigation of a particular problem; when this
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