COMPUTERS IN ALGEBRA:
NEW ANSWERS, NEW QUESTIONS
Cheryl E. Praeger
Abstract. The use and development of computer technology by
algebraists over the last forty years has revolutionised the way in
which algebraists think about algebra, and the way they teach it
and conduct their research....
J. Korean Math. Soc. 38 (2001), No. 4, pp. 763–781
COMPUTERS IN ALGEBRA:
NEW ANSWERS, NEW QUESTIONS
Cheryl E. Praeger
Abstract. The use and development of computer technology by
algebraists over the last forty years has revolutionised the way in
which algebraists think about algebra, and the way they teach it
and conduct their research. This paper is a personal reflection on
these changes by a somewhat unwilling computer user.
1. Introduction
This paper is a personal reflection by a somewhat reluctant computer
user on the crucial role played by the computer in algebra research over
the past thirty years, and its likely future importance. Thirty years is
the extent of my personal knowledge, but computers have been used in-
novatively in algebra for more than forty years, and accounts of the very
early period can be found in [43, 57]. First came answers to several math-
ematical questions which demonstrated the power of computers. This
served to raise new questions thereby inspiring established researchers,
young mathematicians and computer scientists to channel their energies
into designing new algorithms and new computer algebra systems. As
a result the number-crunching devices of the 1970’s were transformed
into sophisticated oracles which seem almost to understand the way an
algebraist thinks.
This metamorphosis in technology occurred in parallel with an equally
dramatic change in the way algebraists think about their subject and the
way they conduct their research. On the one hand computers facilitate
mathematical discovery, and on the other hand they can be integral to
proving theorems. It is impossible to overemphasise the impact of com-
puter systems such as Magma [12] and GAP [54] on the professional
Received October 19, 2000.
1991 Mathematics Subject Classification: 20B40, 20C40.
Key words and phrases: computational algebra, computational group theory.
,764 Cheryl E. Praeger
lives of algebraists internationally, both on their teaching and on their
research. All of my research students use these computer systems as es-
sential learning tools to explore new concepts. They demand illustrative
examples to examine by computer to aid their understanding.
Answers to mathematical questions raised new questions which in
turn inspired new conceptual breakthroughs. Computers have become
an experimental tool for exploring new concepts and structures in al-
gebra, for spotting patterns, and for suggesting new conjectures and
theorems. Demand for higher performance and greater mathematical ca-
pabilities for investigating larger and more complex mathematical struc-
tures was the impetus for new algorithm development. This in turn
highlighted the importance of complexity analysis and statistical analy-
sis for understanding the performance of algorithms. Moreover the use
of computers in algebra, and the mathematics developed to support it,
have led to new areas of algebraic research which integrate and build on
diverse areas of mathematics. Computers have become an indispensable
tool at the forefront of cutting-edge research in algebra.
Because of my knowledge and experience the paper will focus on group
theoretic illustrations rather than examples from other areas of alge-
bra. Also, because I will concentrate on areas with which I have had
some personal involvement I will make very little mention of several
important lines of development related to computational group theory,
such as computation with finitely presented groups [10] and crystallo-
graphic groups [13], computation with characters and representations of
groups [23, 44], and computational complexity [1, 6, 37].
2. Computers as labour saving devices
My initial undergraduate education did not include any computer
courses. There were none available. My first introduction to computers
was a short course on the then new computer language Fortran IV at the
Australian National University in December 1968. I was spending my
summer at the Australian National University on a Vacation Scholarship
at the end of my third year of undergraduate study. I found the Fortran
language interesting, but the computer interface tedious and irritating.
The only mathematically significant applications suggested as exercises
during the course were two procedures to generate prime numbers. Our
programs were presented to the computer in a stack of punched cards,
, Computers in algebra: new answers, new questions 765
and we could have our cards run through the computer once each day.
I decided that an application which needed the use of computers would
have to be very important to me to warrant the tedium of daily de-
bugging of a stack of punched cards.
As a research student in Oxford in the early 1970’s I recognised the
value of computers as labour saving devices for performing tasks that I
could easily see how to do by hand but which could be done faster when
automated. Examples of this were procedures for finding all solutions for
certain divisibility conditions or linear inequalities over some restricted
range of parameter values. My D. Phil. supervisor Peter Neumann
had an ongoing ‘spare-time’ research program to classify the primitive
permutation groups of prime degree less than 100. One purpose of this
exercise was to test the power of the existing theory of permutation
groups, and another purpose no doubt was to sharpen the understanding
and expertise of his research students by involving them in some of the
cases. It was good fun working together with Peter on this programme.
I remember a group of us going to the University’s computer centre
to submit our program on punched cards. We would wait until our
allotted 10 seconds of CPU time was used up, and re-cycle the cards
for another run with the values of the parameters adjusted slightly. It
was an enjoyable social experience, but I did not regard it as serious
mathematical work.
3. Permutation groups and simple groups
The first time I experienced a sense of awe at what could be achieved
in algebra using computers was in 1973. I was attending a course of
lectures by Charles Sims in Oxford. The highlight of these lectures was
Sims’ description of his construction [56] of the Lyons-Sims sporadic
simple group of order
51, 765, 179, 004, 000, 000 = 28 · 37 · 56 · 7 · 11 · 31 · 37 · 67
as a group of permutations of a set of size 8,835,156. Work of Richard
Lyons [38] predicted that such a group might exist, but although a great
deal could be worked out theoretically about its structure, without an
explicit construction, neither its existence nor its uniqueness could be
proved.
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