# A BOOK OF ABSTRACT ALGEBRA PINTER

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A BOOK OF. ABSTRACT ALGEBRA. Second Edition. Charles C. Pinter. Professor of Mathematics. Bucknell University. Dover Publications, Inc., Mineola, New. Buy A Book of Abstract Algebra: Second Edition (Dover Books on Mathematics) Charles C. Pinter is Professor Emeritus of Mathematics at Bucknell University. A Book of Abstract Algebra book. Read 10 reviews from the world's largest community for readers.

A Book of Abstract Algebra: Second Edition. Charles C Pinter. Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises.

If it checks, then e is an identity element.

For iv , first note that if there is no identity element, there can be no inverses. If this checks, x' is the inverse of x.

Therefore, — 1 is the identity element. Therefore, — x — 2 is the inverse of x. Every element has an inverse. Use the format explained on page Label these operations 0 l to 0 Remember that, in general, there are other possible operations on G, so it may not always be clear which is the group's oper- ation unless we indicate it.

If there is no danger of confusion, we shall denote the group simply with the letter G. The groups which come to mind most readily are found in our familiar number systems. Here are a few examples. The set Z, with the operation of addition, is obviously a group.

Mostly, we denote it simply by the symbol Z. Most often we denote it simply by Q. The symbol IR represents the set of the real numbers. Groups occur abundantly in nature. This statement means that a great many of the algebraic structures which can be discerned in natural phe- nomena turn out to be groups. Typical examples, which we shall examine later, come up in connection with the structure of crystals, patterns of symmetry, and various kinds of geometric transformations.

Groups are also important because they happen to be one of the fundamental building blocks out of which more complex algebraic structures are made. Especially important in scientific applications are the finite groups, that is, groups with a finite number of elements. It is not surprising that such groups occur often in applications, for in most situations of the real world we deal with only a finite number of objects.

The easiest finite groups to study are those called the groups of integers modulo n where n is any positive integer greater than 1. These groups will be described in a casual way here, and a rigorous treatment deferred until later.

## Book of Abstract Algebra

Let us begin with a specific example, say, the group of integers modulo 6. Imagine the numbers 0 through 5 as being evenly distributed on the circumference of a circle. To add two numbers h and k, start with h and o 3 move clockwise k additional units around the circle: From geometrical considerations it is clear that this kind of addition by suc- cessive rotations on the unit circle is associative.

If the commutative law holds in a group G, such a group is called a com- mutative group or, more commonly, an abelian group. Abelian groups are named after the mathematician Niels Abel, who was mentioned in Chapter 1, and who was a pioneer in the study of groups. Let G be the group which consists of the six matrices -G: This group has the following operation table, which should be checked: The details are simple. For example, B is the inverse of D, A is the inverse of A, and so on. Thus, G is a group!

Examples of Abelian Groups Prove that each of the following sets, with the indicated operation, is an abelian group. Proceed as in Chapter 2, Exercise B. U x U may therefore be identified with the set of all the points in the plane. Which of the following subsets of R x R, with the indicated operation, is a group? Which is an abelian group? Proceed as in the preceding exercise. Is this a group? It is also associative, as the accompanying pictorial representation suggests: Let the union of A, B, and C be divided into seven regions as illustrated.

Prove the following: List the elements of P D. Do not forget the empty set and the whole set D. There is a single checker on the board, and it has four possible moves: Move vertically; that is, move from 1 to 3, or from 3 to 1, or from 2 to 4, or from 4 to 2.

## A Book of Abstract Algebra

Move horizontally, that is, move from 1 to 2 or vice versa, or from 3 to 4 or vice versa. Move diagonally, that is, move from 2 to 3 or vice versa, or move from 1 to 4 or vice versa. Stay put. We may consider an operation on the set of these four moves, which consists of performing moves successively. For example, if we move horizontally and then vertically, we end up with the same result as if we had moved diagonally: And so on. In this game there are eight possible moves: Flip over the coin at A.

Flip coin at A; then switch. Flip over the coin at B. Flip coin at B; then switch. Flip over both coins. Flip both coins; then switch. Switch the coins. Do not change anything. Is it commutative? If not, show why not. Groups in Binary Codes The most basic way of transmitting information is to code it into strings of Os and Is, such as , , etc. Such strings are called binary words, and the number of 0s and Is in any binary word is called its length.

All information may be coded in this fashion. When information is transmitted, it is sometimes received incorrectly. One of the most important purposes of coding theory is to find ways of detecting errors, and correcting errors of transmission.

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*PAINLESS ALGEBRA PDF*

We will prove that the operation of word addition has the following properties on B": It is commutative. It is associative. There is an identity element for word addition. Every word has an inverse under word addition. First, we verify the commutative law for words of length 1: Show that a 1 ,a 2 , To verify the associative law, we first verify it for words of length 1: The identity element of B", that is, the identity element for adding words of length n, is: The inverse, with respect to word addition, of any word a lt Well, suppose e 1 and e 2 are identity elements of some group G.

Can an element a in a group have two different inverses'! This shows that in every group, each element has exactly one inverse. Remember that a and b do not have to be numbers and therefore " sum " does not, in general, refer to adding numbers. Once again, remember that "product" does not, in general, refer to multiplying num- bers.

Multiplicative notation is the most popular because it is simple and saves space. In the remainder of this book multiplicative notation will be used except where otherwise indicated. In particular, when we represent a group by a letter such as G or H, it will be understood that the group's operation is written as multiplication.

There is common agreement that in additive notation the identity el- ement is denoted by 0, and the inverse of a is written as — a. It is called the negative of a. In multiplicative notation the identity element is e and the inverse of a is written as aT 1 "a inverse". This will be our first theorem about groups. This is the idea of the proof; now here is the proof: Why not? This theorem tells us that if the product of two elements is equal to e, these elements are inverses of each other.

In particular, if a is the inverse of b, then b is the inverse of a. The next theorem gives us important information about computing inverses.

The next formula tells us that a is the inverse of the inverse of a. The proof of i is as follows: The proof of ii is analogous but simpler: The associative law states that the two products a bc and ab c are equal; for this reason, no confusion can result if we denote either of these products by writing abc without any parentheses , and call abc the product of these three elements in this order. The net effect of the associative law is that parenth- eses are redundant.

In the exercises below, the exponential notation a" is used in the following sense: Solving Equations in Groups Let a, b, c, and x be elements of a group G. In each of the following, solve for x in terms of a, b, and c. Rules of Algebra in Groups For each of the following rules, either prove that it is true in every group G, or give a counterexample to show that it is false in some groups.

This is the same as saying that every element of G has a "square root. Elements which Commute If a and b are in G and ab — ba, we say that a and b commute. Assuming that a and b commute, prove the following: The abbreviation iff stands for "if and only if. Proceed roughly as in Exercise Set A. Let a, b, c denote elements of G, and let e be the neutral element of G.

When the exercises in a Set are related, with some exercises building on pre- ceding ones, so they must be done in sequence, this is indicated with a symbol t in the margin to the left of the heading. See Theorem 2. Then ba is the inverse of ab. Counting Elements and Their Inverses Let G be a finite group, and let S be the set of all the elements of G which are not equal to their own inverse.

The set S can be divided up into pairs so that each element is paired off with its own inverse. Constructing Small Groups In each of the following, let G be any group. Let e denote the neutral element of G. Suppose x appears twice in the row of a: Indeed, keeping in mind parts 1 and 2 above, there is only one way of completing the following table.

Do so! You need not prove associativity. Using only part 1, above, complete the following group table of G. Complete the group table of G, as in the preceding exercise. Direct Products of Groups If G and H are any two groups, their direct product is a new group, denoted by G x H, and defined as follows: CxH consists of all the ordered pairs x, y where x is in G and y is in H.

The identity element of G x HI is: G3 For each a, b e G x H, the inverse of a, b is: There are six elements, each of which is an ordered pair. The notation is additive.

Every element of the group is its own inverse. Prove that G x H also has this property. Prove by induction. You may conclude that it is true for every positive integer n. Try a 2. It may happen though it doesn't have to that the product of every pair of elements of S is in S.

If it happens, we say that S is closed with respect to multiplication. Then, it may happen that the inverse of every element of S is in S. In that case, we say that S is closed with respect to inverses. If both these things happen, we call S a subgroup of G. If the negative of every element of S is in S, we say that S is closed with respect to negatives. If both these things happen, S is a subgroup of G. Indeed, the sum of any two even integers is an even integer, and the negative of any even integer is an even integer.

Furthermore, the product of any two rational numbers is rational, and the inverse that is, the reciprocal of any rational number is a rational number. An important point to be noted is this: In other words, if a and b are elements of S, the product ab computed in S is precisely the product ab computed in G. The importance of the notion of subgroup stems from the following fact: It is easy to see why this is true.

To begin with, the operation of G, restricted to elements of S, is certainly an operation on S. It is associative: Next, the identity element e of G is in S and continues to be an identity element in S: Finally, every element of S has an inverse in S because S is closed with respect to inverses. Thus, S is a group! One reason why the notion of subgroup is useful is that it provides us with an easy way of showing that certain things are groups.

Indeed, if G is already known to be a group, and S is a subgroup of G, we may conclude that S is a group without having to check all the items in the definition of "group. Many of the groups we use in mathematics are groups whose elements are functions. In fact, historically, the first groups ever studied as such were groups of functions.

In calculus we learned how to add functions: The details are simple, but first, let us remember what it means for two functions to be equal. At the other extreme, the whole group G is obviously a subgroup of itself.

They are called the trivial subgroups of G. All the other subgroups of G are called proper subgroups. Suppose G is a group and a, b, and c are elements of G. Define S to be the subset of G which contains all the possible products of a, b, c, and their inverses, in any order, with repetition of factors permitted. Thus, typical elements of S would be abac" 1 , c" 1 a" i bbc, and so on.

It is easy to see that S is a subgroup of G: S is called the subgroup of G generated by a, b, and c. In particular, if a is a single element of G, we may consider the subgroup generated by a. The student should check the table to verify this. For example, the additive group Z 6 is cyclic. What is its generator?

Every finite group G is generated by one or more of its elements obvi- ously. A set of equations, involving only the generators and their inverses, is called a set of defining equations for G if these equations completely determine the multiplication table of G. For example, to find the product of ab and ab 2 , we compute as follows: All the entries in the table of G may be computed in the same fashion.

When a group is determined by a set of generators and defining equa- tions, its structure can be efficiently represented in a diagram called a Cayley diagram. These diagrams are explained in Exercise G. Recognizing Subgroups In each of the following, determine whether or not H is a subgroup of G. Assume that the operation of H is the same as that of G.

## Full text of "Charles C. Pinter — A Book of Abstract Algebra"

If H is a subgroup of G, show that both conditions in the definition of "subgroup" are satisfied. If H is not a subgroup of G, explain which condition fails. Note that in this example the operation of G and H is multiplication. In the next problem, it is addition. Use the following formula from trigonometry: See Chapter 3, Exercise C.

Subgroups of Abelian Groups In the following exercises, let G be an abelian group. Prove that H is a subgroup of G. Prove that K is a subgroup of G. Subgroups of an Arbitrary Group Let G be a group. Prove that C is a subgroup of G. Suppose e e S, and that S is closed with respect to multiplication.

Prove that S is a subgroup of G. It remains to prove that G is closed with respect to inverses. Show that Z 3 x Z 4 is a cyclic group. Write the table of G. G is called the quaternion group. Cayley Diagrams Every finite group may be represented by a diagram known as a Cayley diagram. A Cayley diagram consists of points joined by arrows. There is one point for every element of the group. The arrows represent the result of multiplying by a generator.

As another example, the inverse of ab 2 is the path which leads from ab 2 back to e. We note instantly that this is ba. A point-and-arrow diagram is the Cayley diagram of a group iff it has the following two properties: Cayley diagrams are a useful way of finding new groups.

Write the table of the groups having the following Cayley diagrams: You may take any point to represent e, because there is perfect symmetry in a Cayley diagram. Choose e, then label the diagram and proceed. A function is generally defined as follows: If A and B are sets, then a function from A to B is a rule which to every element x in A assigns a unique element y in B. There is nothing inherently mathematical about this notion of function.

For example, imagine A to be a set of married men and B to be the set of their wives. No pun is intended. This notation is useful whenever A is a finite set: For example, if we look at the function immediately above, we observe that a and b both have the same image x.

Thus, Definition 1 A function f: Thus, a convenient definition of "injective" is this: Definition 2 A function f: By Definitions 1 and 2, each element of B is the image of at least one element of A, and no more than one element of A.

So each element of B is the image of exactly one element of A. Definition 3 A function f: The most natural way of combining two functions is to form their "composite.

The composite function de- noted by g of is a function from A to C defined as follows: Let C be the set of all the mothers of members of B. Let us tackle each of these claims in turn. Indeed, the rule "divide by 2" undoes what the rule "multiply by 2" does. To sum up: A function f: In that case, the inverse f' 1 is a bijective function from B to A. Prove your answer in either case. Proof Take any element y e U. Proof By exhibiting a counterexample: Functions on IR and Z Determine whether each of the following functions is or is not a injective, and b surjective.

Proceed as in Exercise A. Functions on Arbitrary Sets and Groups Determine whether each of the following functions is or is not a injective, b surjective. In parts 1 to 3 A and B are sets, and A x B denotes the set of all the ordered pairs x, y as x ranges over A and y over B.

Composite Functions Find the composite function, as indicated. Alternatively, they used a code g which interchanged the letters a with o, i with u, e with y, and s with t. Describe its inverse. Functions on Finite Sets 1 The members of the UN Peace Committee must choose, from among themselves, a presiding officer of their committee. For each member x, let f x designate that member's choice for officer.

Look at part 1. If not, give a counterexample. Is the converse of this statement true: If "yes," prove it; if "no," give a counterexample. In elementary algebra we learned to think of a permutation as a rearrangement of the elements of a set.

It is clear, therefore, that there is no real difference between the new definition of permutation and the old. The new definition, however, is more general in a very useful way since it allows us to speak of permutations of sets A even when A has infinitely many elements. In Chapter 6 we saw that the composite of any two bijective functions is a bijective function. Thus, the composite of any two permutations of A is a permutation of A.

We have just seen that composition is an associative operation. Is there a neutral element for composition? The second equation is proved analogously. We saw in Chapter 6 that the inverse of any bijective function exists and is a bijective function. Thus, the inverse of any permutation of A is a permutation of A. Let us recapitulate: This operation is associative. For any set A, the group of all the permutations of A is called the symmetric group on A, and it is represented by the symbol S A.

Let us take a look at S 3. A graphic way of re- presenting this is: The other combinations of elements of S 3 may be computed in the same fashion. The student should check the following table, which is the table of the group S 3: Among the most interesting groups of permutations are the groups of symmetries of geometric figures.

We will see how such groups arise by considering the group of symmetries of the square. We may think of a symmetry of the square as any way of moving a square to make it coincide with its former position.

Every time we do this, vertices will coincide with vertices, so a symmetry is completely described by its effect on the vertices. The operation on symmetries is composition: The eight symmetries of the square form a group under the oper- ation o of composition, called the group of symmetries of the square.

## A Book of Abstract Algebra: Second Edition

These groups are called the dihedral groups. For example, the group of the square is Z 4 , the group of the pentagon is D 5 , and so on.

Every plane figure which exhibits regularities has a group of sym- metries. Artificial as well as natural objects often have a surprising number of symmetries. Far more complicated than the plane symmetries are the symmetries of objects in space. Modern-day crystallography and crystal physics, for exam- ple, rely very heavily on knowledge about groups of symmetries of three- dimensional shapes. Groups of symmetry are widely employed also in the theory of electron structure and of molecular vibrations.

In elementary particle physics, such groups have been used to predict the existence of certain elementary parti- cles before they were found experimentally! Symmetries and their groups arise everywhere in nature: Examples of Groups of Permutations 1 Let G be the subset of S 4 consisting of the permutations Show that G is a group of permutations, and write its table.

Write its table. List them, then write the table of this group: Show that G is a subgroup of S A , and write the table of G. G has eight elements. List them, and write the table of G. Prove that G is a subgroup of S K. Indicate a generator of G. Show that G is a subgroup of S u.

Symmetries of Geometric Figures 1 Let G be the group of symmetries of the regular hexagon.

List the elements of G there are 12 of them , then write the table of G. List the elements of G there are four of them , and write the table of G.

Do the same for the letters V and H. Assume that the three arms are of equal length, and the three central angles are equal. It is unaltered when the sub- scripts undergo any of the following permutations: The sym- metries of a polynomial p are all the permutations of the subscripts which leave p unchanged. They form a group of permutations.

List the symmetries of each of the following polynomials, and write their group table. Prove that G is a subgroup of S A. Prove that 'G is a subgroup of S A. Because of their practical importance, this chapter will be devoted to the study of a few special properties of permutations of finite sets.

If n is a positive integer, consider a set of n elements. We have already seen that the group of all the permutations of this set is called the symmetric group on n elements and is denoted by S n. One of the most characteristic activities of science any kind of science is to try to separate complex things into their simplest component parts. This intellectual "divide and conquer" helps us to understand complicated processes and solve difficult problems.

The savvy mathematician never misses the chance of doing this whenever the opportunity presents itself. We will see now that every permutation can be decomposed into simple parts called "cycles," and these cycles are, in a sense, the most basic kind of permutations. These closed chains may be considered to be the component parts of the permutation; they are called "cycles. Every permutation breaks down, just as this one did, into separate cycles. Actually, it is very easy to compute the product of two cycles by reasoning in the following manner: Let us continue with the same example, Remember that the permutation on the right is applied first, and the permu- tation on the left is applied next.

For example, is a cycle of length 4. If two cycles have no elements in common they are said to be disjoint. For example, and are disjoint cycles, but and are not disjoint. Disjoint cycles commute: We are now ready to prove what was asserted at the beginning of this chapter: Every permutation can be decomposed into cycles — in fact, into disjoint cycles.

More precisely. Next, we take the first number which hasn't yet been used, namely 2. We are done: Let a! Next, let bi be the first number which has not yet been examined and such that f b 1 b t. Incidentally, it is easy to see that this product of cycles is unique, except for the order of the factors.

Now our curiosity may prod us to ask: Or is there some way of simplifying it further? A cycle of length 2 is called a transposition. It is a fact both remarkable and trivial that every cycle can be expressed as a product of one or more transpositions.

Thus, every permutation, after it has been decomposed into disjoint cycles, may be broken down further and expressed as a product of transpo- sitions. However, the expression as a product of transpositions is not unique, and even the number of transpositions involved is not unique.

Nevertheless, when a permutation n is written as a product of transpo- sitions, one property of this expression is unique: This fact will be proved in a moment. For example, we have just seen that can be written as a product of four transpositions and also as a product of six transpositions; it can be written in many other ways, but always as a product of an even number of transpositions.

Likewise, can be de- composed in many ways into transpositions, but always an odd number of transpositions. A permutation is called even if it is a product of an even number of transpositions, and odd if it is a product of an odd number of transpo- sitions. What we are asserting, therefore, is that every permutation is unam- biguously either odd or even.

Thanks for telling us about the problem. Return to Book Page. Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra.

Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. Intended for undergraduate courses in abstract algebra, it is suitable for junior- and senior-level math Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Intended for undergraduate courses in abstract algebra, it is suitable for junior- and senior-level math majors and future math teachers.

This second edition features additional exercises to improve student familiarity with applications. An introductory chapter traces concepts of abstract algebra from their historical roots. Succeeding chapters avoid the conventional format of definition-theorem-proof-corollary-example; instead, they take the form of a discussion with students, focusing on explanations and offering motivation. Each chapter rests upon a central theme, usually a specific application or use.

The author provides elementary background as needed and discusses standard topics in their usual order. He introduces many advanced and peripheral subjects in the plentiful exercises, which are accompanied by ample instruction and commentary and offer a wide range of experiences to students at different levels of ability.

Get A Copy. Paperback , Second Edition , pages. Published January 14th by Dover Publications first published March 1st More Details Original Title. Other Editions 5. Friend Reviews. To see what your friends thought of this book, please sign up. To ask other readers questions about A Book of Abstract Algebra , please sign up. Be the first to ask a question about A Book of Abstract Algebra.

Lists with This Book. Community Reviews. Showing Rating details. More filters. Sort order. Nov 17, David Mendiola rated it it was amazing.

Holy morphism, this book is great. I can't wait until they inevitably publish the gilded-leaf KJV of it. The first chapter is enough to get you psyched about Abstract Algebra. He makes it perfectly clear what our motivation is, so when groups and rings start coming, they feel so natural rather than like arcane math objects.

I've been evangelizing to friends and coworkers the spirit of algebra with very limited success. If I may be so bold as to skirt on the edges of blasphemy, some other peopl Holy morphism, this book is great. If I may be so bold as to skirt on the edges of blasphemy, some other people might have a fleeting thought that it would possibly, be it Pinter's will, benefit the neophyte to have an ever so slightly extended answer key.

I'm sure there are other great books on abstract algebra, but this is just perfect as an introduction. The conversational style feeds me the material just the way I like it: For Graduate Students and Advanced Undergraduates. Introduction to Proof in Abstract Mathematics.

An Introduction to Linear Algebra and Tensors. Abstract Algebra and Solution by Radicals. Basic Algebra II: Second Edition. Basic Algebra I: Mathematical Physics:

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