The introduction describes this text as one which could be used for either an undergraduate or a graduate course. The text incorporates technology as a means of reducing computation and allowing students to create their own examples. It is written so as to accommodate either Mathematica or a freeware program entitled Groups, Algorithms and Programming (GAP). As an example of how this can simplify life for undergraduates, both Mathematica and GAP can verify (by brute force) that A5 is simple. However, such a brute force analysis doesn’t really help us to see why A5 is a simple group.
Based on my experience, undergraduate students will find the effort required to learn Mathematica or GAP notation to exceed the reward of creating the examples. The MAA software package Exploring Small Groups is much easier to use. Additionally, Excel will create modular arithmetic tables very easily.
The table of contents suggests a traditional order of topics, with group theory being explored in full prior to the introduction of rings. However, the section and chapter headings often don’t match up with what is actually covered. Homework problems are grouped at the end of each chapter. In the chapters I looked closely at, many of the key ideas and examples (e.g., the dihedral groups) appear only in homework problems. This book does not appear to be of sufficient depth or rigor for a graduate text, but the organization of the text makes it inappropriate for undergraduates as well. I outline some of those difficulties below, focusing on four of the chapters.
Chapter One: Understanding the Group Concept
As is common, the notion of a group is first introduced through the symmetries of an equilateral triangle. In this case the author introduces Terry the Dancing (equilateral) Triangle. Terry can perform six dance steps each of which corresponds to one of the six members of . Unfortunately, the pictures illustrating Terry’s six moves do not illustrate results of the dance moves as described. Instead, the only pictures provided are taken from “the animation close to the completion of each step.” As a result a rotation of 120 degrees clockwise is illustrated by a figure which appears to have been rotated 60 degrees. The figures illustrating the three reflections show triangles with are no longer equilateral! I think the reader deserves a set of pictures of Terry after he has completed each step. Without that it is not possible to verify the group table that the author presents.
The result is that what could have been a very clever introduction is confusing. The confusion quickly deepens. After stating each of the group axioms in terms of dance steps (STAY is the name used for the “identity dance step.”) At this point the group axioms have not yet been stated in any other terms, only described as dance steps, but we immediately proceed to the following proposition
If y is an inverse of x, then x is the only inverse of y.
While the Proposition as stated is true, I’m almost certain it was meant to be stated as: “if x is an inverse of y then x is the only inverse of y.” At least that’s what its “proof” proves!
The next example considered is modular arithmetic. The first example is addition mod 10, but the notation used is a dot. That is a + b (Mod 10) is symbolized by a.b. As a result, we see (on page 7) the equation x.0 = 0.x = x. While true in this context, it is likely to be very confusing to beginning students. Part of the confusion is due to the Mathematica syntax which uses MultTable for any Cayley Table for a group, so that any group operation is “multiplication.”
The chapter also contains proofs of several facts from elementary number theory. Several of these proofs use the well-ordering of the positive integers, even though this theorem is never explicitly stated. I think these concepts are better reserved for a “chapter 0” (as Gallian and others do in their texts). The chapter concludes with the formal definition of a group. It does not, in my opinion, have enough examples of groups to give beginning students an idea of the richness of the group concept.
Chapter Two: The Structure within a Group
There is way too much information here for an undergraduate course. Among the topics (covered in 20 pages) are: generators, subgroups, cyclic groups, the group of units mod n, and some more number theory. At the undergraduate level, each of these concepts is probably worth a chapter in its own right. Here are examples of how condensed things are:
- The subgroup structure of cyclic groups is explored only in the homework.
- The group of units in Zn is defined in the middle of a paragraph in chapter 1. Some 16 pages later, Euler’s totient function is proved to count the number of generators of Zn and an example of the group of units is given.
- S3 is defined, but there is no reference to the fact that this is part of a family of groups which will appear again in Chapter 5.
Chapter 4: Mappings between Groups
Here again, the material is highly condensed. The motivating example of an isomorphism is between a quotient group of the octahedral group and S3! After defining isomorphism, the author proceeds to an enumeration, with proofs, of all the groups (up to isomorphism) of order 8. D4 appears, of course — but the only time we have previously encountered this group is as problem 6 in chapter 1. In fact, there is no discussion of the dihedral groups to this point outside of D3 and D4. I can’t imagine a typical undergraduate understanding, much less appreciating what’s going on here. Why do we care how many groups of order 8 there are? The author then proceeds to proofs of the three classic isomorphism theorems. The only examples of these are in the exercises. I think undergraduates need lots more examples of actual isomorphisms before we lay all this theory on them.
Chapter 9: Introduction to Rings
The first section in this chapter begins by listing a few sets which admit two binary operations. We then leave ring theory altogether for a brief excursion into Cantor’s set theory! The author defines a countable set and proves that the rational numbers are countable and that the real numbers are not. He then returns to the idea of two binary operations without any explanation as to what the previous few pages were about. The first example of a ring is Z6, the second is the quaternions. There is a discussion of zero divisors and the fact that elements which are not zero divisors don’t always have inverses, but the notion of the group of units in Zn is not discussed. A great deal of space is devoted to the syntax needed to enter ring definitions into Mathematica and GAP — the result is that there are far fewer examples than I would like to see in an initial treatment of the ring concept.
At this point, I think it fair to conclude that this text is not competitive with the many fine undergraduate and graduate texts available. I would not recommend this book as either an undergraduate or a graduate text. It is too condensed for an undergraduate text and doesn’t provide enough depth for a graduate text. In addition, the presentation style is not conducive to student learning.
While the use of Mathematica or GAP has great potential, it is not sufficient to overcome the deficiencies I observed in these five chapters. In fact, the numerous explanations of the use of Mathematica and GAP obscure the presentation. I think these would be more helpful as an appendix or a separate manual.
Richard Wilders is Marie and Bernice Gantzert Professor in the Liberal Arts and Sciences and Professor of Mathematics at North Central College. His primary areas of interest are the history and philosophy of mathematics and of science. He has been a member of the Illinois Section of the Mathematical Association of America for 30 years and is a recipient of its Distinguished Service Award.