00319 - Problems in Group Theory [Dixon].pdf

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PROBLEMS
IN GROUP
THEORY
A BLAISDELL BOOK IN
CONSULTING
EDITOR
PURE AND APPLIED
MATHEMATICS
Garrett Birkhoff,
Harvard University
John D. Dixon
UNIVERSITY OF NEW SOUTH WALES
Probleins
in Group
Theory
BLAISDELL PUBLISHING COMPANY
A Division of Ginn and Company
WALTHAM, MASSACHUSETTS
•
TORONTO
•
LONDON
COPYRIGHT
©
1967
BY BLAISDELL PUBLISHING COMPANY,
A DIVISION OF GINN AND COMPANY.
ALL RIGHTS RESERVED.
LIBRARY OF CONGRESS CATALOG CARD NUMBER:
PRINTED IN THE UNITED STATES OF AMERICA.
67-10993
Preface
Group theory is a basic branch of algebra. The simple definition of a
group, as expressed through the group axioms, together with the way that
group theory interpenetrates many other fields in mathematics, make it
a good introduction to an advanced study of many areas of mathematics.
In the century and a half since the foundations were laid, many great
mathematicians have become interested in the study of group theory
because its elementary nature scarcely conceals its many deep and tanta-
lizing problems. Like number theory, group theory has many problems
which may be stated in simple terms but whose solutions lie very deep.
Recent activity in group theory has resulted in the solution of several
long-standing problems. Even a complete classification of all finite simple
groups (the basic units from which all finite groups are built) now appears
possible, in a large part due to the work of the mathematician John G.
Thompson. On one hand, there has been the discovery and development
of new tools applicable to old problems, and on the other, we have dis-
covered how the older methods can be applied to new and interesting
problems. The survey articles of Brauer [27] and Wielandt and Huppert
[66] illustrate these developments.
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