This book began about 20 years ago in the form of supplementary notes for my
algebra classes. I wanted to discuss some concrete topics such as symmetry, linear
groups, and quadratic number fields in more detail than the text provided, and to
shift the emphasis in group theory from permutation groups to matrix groups.
Lattices, another recurring theme, appeared spontaneously. My hope was that the
concrete material would interest the students and that it would make the abstractions
more understandable, in short, that they could get ferther by learning both at the
same time. This worked pretty well. It took me quite a while to decide what I
wanted to put in, but I gradually handed out more notes and eventually began
teaching from them without another text. This method produced a book which is, I think,
somewhat different from existing ones. However, the problems I encountered while
fitting the parts together caused me many headaches, so I can't recommend starting
this way.
The main novel feature of the book is its increased emphasis on special topics.
They tended to expand each time the sections were rewritten, because I noticed over
the years that, with concrete mathematics in contrast to abstract concepts, students
often prefer more to less. As a result, the ones mentioned above have become major
parts of the book. There are also several unusual short subjects, such as the Todd-
Coxeter algorithm and the simplicity of PSL2.
