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A student in the primary or secondary schools frequently has the notion that mathematics is concerned solely with number and measure.
However, mathematics has always been much more than merely a quantitative science with applications to activities such as bookkeeping and money-changing it is deeply concerned with logic and structure.
The theory of groups is one of the important non-quantitative branches of mathematics. The concept of a group, although comparatively recent
in the development of mathematics, has been most fruitful for example, it has been a powerful tool in the investigation of algebraic equations,
of geometric transformations, and of problems in topology and number theory.
Two features of group theory have traditionally made it advisable to postpone its study until rather late in a student’s mathematical education. First, a high degree of abstractness is inherent in group theoretical ideas, and ability to cope with abstract concepts comes with mathematical maturity. Second, the ways in which group theory interacts with other fields of study to illuminate and advance them can be seen only after long and elaborate development of the theory, and then only by students acquainted with the other fields. In this book we have aimed at a presentation suitable for students at a relatively early stage of mathematical growth.
To bypass the difficulties stemming from abstractness, we have used geometric pictures of groups-graphs of groups.
In this way, abstract groups are made concrete in visual patterns that correspond to group structure. However, we cannot hope to provide a substitute for the prolonged reading and study necessary to grasp the concepts of varied mathematical fields of inquiry. We have tried to make
the best of this situation by indicating the broader significance of some of the theorems and concepts presented.
We acknowledge that we cannot always motivate the reader with “practical” applications. Ultimately, we have to rely on the appeal of the mathematical content in and of itself. Of course, the most effective spur comes from the reader himself this has to be his contribution.
Preface
Introduction to Groups
Group Axioms
Examples of Groups
Multiplication Table of a Group
Generators of a Group
Graph of a Group
Definition of a Group by Generators and Relations
Subgroups
Mappings
Permutation Groups
Normal Subgroups
The Quaternion Group
Symmetric and Alternating Groups
Path Groups
Groups and Wallpaper Designs
Appendix: Group of the Dodecahedron and the Icosahedron
Solutions
Bibliography
Index