Numbers, Groups and Codes

Описание

Introduction
‘A group is a set endowed with a specified binary operation which is associative
and for which there exist an identity element and inverses.’ This, in effect, is
how many books on group theory begin. Yet this tells us little about groups
or why we should study them. In fact, the concept of a group evolved from
examples in number theory, algebra and geometry and it has applications in
many contexts. Our presentation of group theory in this book reflects to some
extent the historical development of the subject. Indeed, the formal definition
of an abstract group does not occur until the fourth chapter. We believe that,
apart from being more ‘honest’ than the usual presentation, this approach has
definite pedagogic advantages. In particular, the student is not presented with
a seemingly unmotivated abstract definition but, rather, sees the sense of the
definition in terms of the previously introduced special cases. Moreover, the
student will realise that these concepts, which may be so glibly presented,
actually evolved slowly over a period of time.
The choice of topics in the book is motivated by the wish to provide a sound,
rigorous and historically based introduction to group theory. In the sense that
complete proofs are given of the results, we do not depart from tradition. We
have, however, tried to avoid the dryness frequently associated with a rigorous
approach. We believe that by the overall organisation, the style of presentation
and our frequent reference to less traditional topics we have been able to overcome this problem. In pursuit of this aim we have included many examples
and have emphasised the historical development of the ideas, both to motivate
and to illustrate. The choice of applications is directed more towards ‘finite
mathematics’ and computer science than towards applications arising out of
the natural sciences.
Group theory is the central topic of the book but the formal definition of a
group does not appear until the fourth chapter, by which time the reader will

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