This second edition covers essentially the same topics as the first.
However, the presentation of the material has been extensively revised
and improved. In addition, there are two new chapters, one dealing with
the fundamental theorem of finitely generated abelian groups and the
other a brief introduction to semigroup theory and automata.
This book is appropriate for second to fourth year undergraduates. In
addition to the material traditionally taught at this level, the book
contains several applications: Polya-Burnside Enumeration, Mutually
Orthogonal Latin Squares, Error-Correcting Codes, and a classification
of the finite groups of isometries of the plane and the finite rotation
groups in Euclidean 3-space, semigroups and automata. It is hoped that
these applications will help the reader achieve a better grasp of the
rather abstract ideas presented and convince him/her that pure
mathematics, in addition to having an austere beauty of its own, can be
applied to solving practical problems.
Considerable emphasis is placed on the algebraic system consisting of
the congruence classes mod n under the usual operations of addition
and multiplication. The reader is thus introduced -- via congruence
classes -- to the idea of cosets and factor groups. This enables the
transition to cosets and factor objects to be relatively painless.
In this book, cosets, factor objects and homomorphisms are introduced
early on so that the reader has at his/her disposal the tools required
to give elegant proofs of the fundamental theorems. Moreover,
homomorphisms play such a prominent role in algebra that they are used
in this text wherever possible.
