Contemporary Abstract Algebra
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Contemporary Abstract Algebra

Joseph A. Gallian

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eBook - ePub

Contemporary Abstract Algebra

Joseph A. Gallian

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About This Book

Contemporary Abstract Algebra, Tenth Edition

For more than three decades, this classic text has been widely appreciated by instructors and students alike. The book offers an enjoyable read and conveys and develops enthusiasm for the beauty of the topics presented. It is comprehensive, lively, and engaging.

The author presents the concepts and methodologies of contemporary abstract algebra as used by working mathematicians, computer scientists, physicists, and chemists. Students will learn how to do computations and to write proofs. A unique feature of the book are exercises that build the skill of generalizing, a skill that students should develop but rarely do. Applications are included to illustrate the utility of the abstract concepts.

Examples and exercises are the heart of the book. Examples elucidate the definitions, theorems, and proof techniques; exercises facilitate understanding, provide insight, and develop the ability to do proofs. The exercises often foreshadow definitions, concepts, and theorems to come.

Changes for the tenth edition include new exercises, new examples, new quotes, and a freshening of the discussion portions. The hallmark features of previous editions of the book are enhanced in this edition. These include:

  • A good mixture of approximately 1900 computational and theoretical exercises, including computer exercises, that synthesize concepts from multiple chapters
  • Approximately 300 worked-out examples from routine computations to the challenging
  • Many applications from scientific and computing fields and everyday life
  • Historical notes and biographies that spotlight people and events
  • Motivational and humorous quotations
  • Numerous connections to number theory and geometry

While many partial solutions and sketches for the odd-numbered exercises appear in the book, an Instructor's Solutions Manual written by the author has comprehensive solutions for all exercises and some alternative solutions to develop a critical thought and deeper understanding. It is available from CRC Press only. The Student Solution Manual has comprehensive solutions for all odd-numbered exercises and many even-numbered exercises.

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Introduction to Groups

And symmetry is a powerful guiding principle that has been used in creating these models [for quantum physics]. The more symmetrical a model is, the easier it is to analyze.
Edward Frenkel, Love and Math
Symmetry is a vast subject, significant in art and nature. Mathematics lies at its root, and it would be hard to find a better one on which to demonstrate the working of the mathematical intellect.
Hermann Weyl, Symmetry

Symmetries of a Square

Suppose we remove a square region from a plane, move it in some way, then put the square back into the space it originally occupied. Our goal in this chapter is to describe all possible ways in which this can be done. More specifically, we want to describe the possible relationships between the starting position of the square and its final position in terms of motions. However, we are interested in the net effect of a motion, rather than in the motion itself. Thus, for example, we consider a 90 rotation and a 450 rotation as equal, since they have the same net effect on every point. With this simplifying convention, it is an easy matter to achieve our goal.
To begin, we can think of the square region as being transparent (glass, say), with the corners marked on one side with the colors blue, white, pink, and green. This makes it easy to distinguish between mo­tions that have different effects. With this marking scheme, we are now in a position to describe, in simple fashion, all possible ways in which a square object can be repositioned. See Figure 1.1. We now claim that any motion—no matter how complicated—is equivalent to one of these eight. To verify this claim, observe that the final position of the square is completely determined by the location and orientation (i.e., face up or face down) of any particular corner. But, clearly, there are only four locations and two orientations for a given corner, so there are exactly eight distinct final positions for the corner.
Figure 1.1
Figure 1.1 Symmetries of a square.
Let’s investigate some consequences of the fact that every motion is equal to one of the eight listed in Figure 1.1. Suppose a square is repositioned by a rotation of 90 followed by a flip about the horizontal axis of symmetry.
Thus, we see that this pair of motions—taken together—is equal to the single motion D. This observation suggests that we can compose two motions to obtain a single motion. And indeed we can, since the eight motions may be viewed as functions from the square region to itself, and as such we can combine them using function composition.
With this in mind, we write H R90 = D because in lower level math courses function composition fg means “g followed by f.” The eight motions R0, R90, R180, R270, H, V, D, and D, together with the operation composition, form a mathematical system called the dihedral group oforder 8 (the order of a group is the number of elements it contains). It is denoted by D4. Rather than introduce the formal definition of a group here, let’s look at some properties of groups by way of the example D4.
To facilitate future computations, we construct an operation table, the Cayley table (so named in honor of the prolific English mathematician Arthur Cayley,bio]Cayley, Arthur who first introduced them in 1854) for D4 below. The circled entry represents the fact that D = HR90. (In general, ab denotes the entry at the intersection of the row with a at the left and the column with b at the top.)
Notice how orderly this table looks! This is no accident. Perhaps the most important feature of this table is that it has been completely filled in without introducing any new motions. Of course, this is because, as we have already pointed out, any sequence of motions turns out to be the same as one of these eight. Algebraically, this says that if A and B are in D4, then so is AB. This property is called closure, and it is one of the requirements for a mathematical system to be a group. Next, notice that if A is any element of D4, then AR0=R0A=A. Thus, combining any element A on either side with R0 yields A back again. An element R0 with this property is called an identity, and every group must have one. Moreover, we see that for each element A in D4, there is exactly one element B in D4 such that AB=BA=R0. In this case, B is said to be the inverse of A and vice versa. For example, R90 and R270 are inverses of each other, and H is its own inverse. The term inverse is a descriptive one, for if A and B are inverses of each other, then B “undoes” whatever A “does,” in the sense that A and B taken together in either order produce R0, representing no change. Another striking feature of the table is that every element of D4 appears exactly once in each row and column. This feature is something that all groups must have, and, indeed, it is quite useful to keep this fact in mind when constructing the table in the first place.
Another property of D4 deserves special comment. Observe that HD DH but R90R180=R180R90. Thus, in a group, ab may or may not be the same as ba. If it happen...

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