Note to Students
This book may be unlike other mathematics textbooks you have read or used in prev ious courses.
The investigations contained in it are designed to facilitate your learning by inviting you to be a n
active participant in the learning process. This is a book that is not meant to be simply read, but
rather engage d. It includes numerous activities within the text that are intended to motivate new
material, illustrate deﬁnitions and theorems, and help you develop both th e intuition and rigor that
is necessary to understand and ap ply ideas from abstract algeb ra.
As professors of mathem atics, we have found (and research conﬁrms) that m a thematics is not
a spectato r sport. To learn and understand mathematics, one must engage in the process of doing
mathematics. This kind of engagement can be cha llenging and even frustrating at times. But if you
are up to th e challenge and willing to take responsibility for your own le arning, yo u will indeed
learn a gr e at deal.
Obviously, this is a book about abstract algebra, and you will learn more about what that means
as we begin our investigations. Our goal, however, is that you will not only learn about abstract
algebra, but that you will also develop a deeper understanding of what mathematics is, how mathe-
matics is done, and how mathematicians think. We hope that you will see that the way mathematics
is developed is often different than how it is pre sented; that d eﬁnitions, theorems, and proofs do not
simply appe ar fully formed in the minds of mathematicians; that math e matical ideas a re highly in-
terconne cted; and that even in a ﬁeld like abstra c t algebra, there is a considerable amount of intuition
to be found.
Thank yo u for joining us on this journey. We hope you enjoy both the challenges and the rewards
that await you in these pages.
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