feature of this text, in that it threads throughout most of the book. . Abstract algebra had its beginnings in attempts to address mathematical problems such as. This book is an introduction to modem (abstract) algebra for undergraduates. The first six available at the book's Web site caite.info Once, when I was a student struggling to understand modern algebra, I was told to view this subject as an intellectual chess game, with conventional moves and.

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I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write this book. I'm sorry that he did not live to see it finished. Arthur was. This book's organizing principle is the interplay between groups and rings, .. in Abstract Algebra, then the prerequisites for this book would be plain. But this is. subject of abstract algebra and no student should go through such a The open source version of this book has received support from the.

Burnside — that every finite, non-abelian, simple group has even order the proof is more than pages long! The manipulation of logical propositions by means of boolean algebra is now called the propositional calculus. This group is isomorphic to D3 and S3. In Figure 2. Design a voting machine for this committee and simplify it as much as possible.

In actual practice, the wiring diagram may not look at all like Figure 2. Two circuits C1 and C2 involving the switches A, B,. Wiring diagram of the circuit. It can be verified that all the axioms for a boolean algebra are valid when interpreted as series-parallel switching circuits.

For example, Figure 2. The zero corresponds to a circuit that is always open, and the unit corresponds to a circuit that is always closed. The complement C of a circuit C is open whenever C is closed and closed when C is open. Thus, the first few primes are 2, 3, 5, 7, 11,. A fundamental fact about P is the prime factorization theorem: See Appendix 2 for a proof of the prime factorization theorem.

Similarly, m is the unique integer in P that is a multiple of both a and b, and is a divisor of every such common multiple. With this background, we can describe some new examples of boolean algebras. It is clear that gcd and lcm are commutative binary operations on Dn , and it is easy to verify that the zero is 1 and the unit is n. Hence the first distributive law holds; the other distributive law and the associative laws are verified similarly.

Thus Dn , gcd, lcm satisfies all the axioms for a boolean algebra except for the existence of a complement. But complements need not exist in general: Hence 6 has no complement, so D18 is not a boolean algebra.

However, all is not lost. The interpretations of the various boolean algebra terms are given in Table 2. If the boolean algebra is the algebra of subsets of X, this relation is the usual inclusion relation. The interpretation of the partial order in various boolean algebras is given in Table 2. A partial order on a finite set K can be displayed conveniently in a poset diagram in which the elements of K are represented by small circles.

The partial order relation is divisibility, so that there is an upward path from a to b if and only if a divides b. Poset diagram of D A boolean algebra is an extremely special kind of poset. We now determine conditions which ensure that a poset is indeed a boolean algebra. It follows from the antisymmetry of the partial order relation that each pair of elements a and b can have at most one greatest lower bound and at most one least upper bound. A lattice is a partially ordered set in which every two elements have a greatest lower bound and a least upper bound.

We can now give an alternative definition of a boolean algebra in terms of a lattice: It can be verified that this definition is equivalent to our original one. In Figure 2. We note in passing that the discussion preceding Example 2. For further reading on lattices in applied algebra, consult, Davey and Priestley [16] or Lidl and Pilz [10].

Lattice that is not a boolean algebra. Poset that is not a lattice. In other words, we would like to reduce this boolean expression to a simpler form.

In actual practice, it is usually desirable to reduce the circuit to the one that is cheapest to build, and the form this takes will depend on the state of the technology at the time; however, for our purposes we take the simplest form to mean the one with the fewest switches. It is difficult to find the simplest form for circuits with many switches, and there is no one method that will lead to that form.

However, we do have methods for determining whether two boolean expressions are equivalent. We can reduce the expressions to a certain normal form, and the expressions will be the same if and only if their normal forms are the same.

We shall look at one such form, called the disjunctive normal form. In the boolean algebra of subsets of a set, every subset can be expressed as a union of singleton sets, and this union is unique to within the ordering of the terms. We shall obtain a corresponding result for arbitrary finite boolean algebras. The elements that play the role of singleton sets are called atoms.

This implies that the atoms are the elements immediately above the zero element in the poset diagram. We now give a more precise description of the algebra of switching circuits.

The atoms of the algebra and the disjunctive normal form of an expression will become clear from this description. An n-variable switching circuit can be viewed as a black box containing n independent switches A1 , A2 ,. The effect of such a circuit can be tested by trying all the 2n different combinations of the n switches and observing when the box allows current to pass. In this way, each circuit defines a function of n variables A1 , A2 ,.

Two circuits give rise to the same switching function if and only if they are equivalent. For example, the circuit in Figure 2.

A1 A2 Figure 2. Each of the 2n elements in the domain of such a function can be mapped to either of the two elements in the codomain. Therefore, the number of different n-variable switching functions, and hence the number of different circuits with n n switches, is Let f and g be the switching functions of two circuits of the n-variables A1 , A2 ,.

The switching function of the opposite circuit to that defining f is f , where f A1 ,. Theorem 2. The zero element is the function whose image is always 0, and the unit element is the function whose image is always 1. This function is the exclusive OR function or a modulo 2 adder. The atoms of Fn are therefore the functions whose image contains precisely one nonzero element. To show that every element of a finite boolean algebra can be written as a join of atoms, we need three preliminary lemmas.

Lemma 2. If A, B1 ,. Poset diagram of the boolean algebra of two-variable switching functions. If B1 ,. The converse implication is trivial. Disjunctive Normal Form. Moreover, this expression is unique up to the order of the atoms.

Therefore, by Lemma 2. Hence there is one atom in the disjunctive normal form for each time the element 1 occurs in the image of the switching function. We see from the values of the switching function in Table 2. We see from Table 2. These atoms partition the Venn diagram in Figure 2. The disjunctive normal form for any boolean expression involving the variables A and B can be calculated by shading the region of the Venn diagram corresponding to the expression and then taking the join of the atoms in the shaded region.

Venn diagram for F2. Venn diagram for F3. Furthermore, the disjunctive normal form provides a method of proving hypotheses derived from these Venn diagrams. However, Venn diagrams become too complicated and impractical for functions of more than four variables. For other general methods of simplifying circuits, consult a book on boolean algebras such as Mendelson [21]. Find the disjunctive normal form and simplify the circuit in Figure 2.

The boolean function f: However, by looking at the Venn diagram in Figure 2. Series-parallel circuit. Venn diagram and simplified circuit. In building a computer, one of the most important pieces of equipment needed is a circuit that will add two numbers in binary form. Consider the problem of adding the numbers 15 and 5. Their binary forms are and , respectively. The binary and decimal additions are shown below. In general, if we add the number.

This is called a half adder. The digits s0 and c1 are functions of a0 and b0 which are given by Table 2.

These circuits are shown in Figure 2. Circuits for the half adder. Suitable expressions for a full adder are as follows. The corresponding circuits are shown in Figure 2. Transistor technology, however, allows us to construct basic switches with multiple inputs. These are called transistor gates. However, any number of inputs is possible. Note that the inversion operation is indicated by a small circle. Transistor gates can be combined in series and in parallel to form more complex circuits.

Any circuit with n inputs and one output defines an n-variable switching function. This is of interest because with certain types of transistors, it is easier to build NOR gates and NAND gates than it is to build the basic operations. Transistor gates. Verify that the circuit in Figure 2. We analyze this circuit by breaking it up into component parts as illustrated in Figure 2. Consider the subcircuit consisting of four NOR gates in Figure 2.

The entire circuit can now be constructed from two of these identical subcircuits together with one NOR gate, as shown in Figure 2. The switching functions for the subcircuit and the full adder are calculated in Table 2. This chip may contain millions of gates and several layers of semiconductor. Simplification of a circuit may not mean the reduction of the circuit to the smallest number of gates.

It could mean simplification to standard modules, or the reduction of the numbers of layers in the chip. In the design of high-speed computers, it is important to reduce the time a circuit will take to perform a given set of operations. However, we now show that every finite boolean algebra is in fact essentially the same as the algebra of subsets of some finite set. To be more precise about what we mean by algebras being essentially the same, we introduce the notion of morphism and isomorphism of boolean algebras.

A morphism between two boolean algebras is a function between their elements that preserves the two binary operations and the unary operation. A boolean algebra isomorphism is a bijective boolean algebra morphism. Isomorphic boolean algebras have identical properties. For example, their poset diagrams are the same, except for the labeling of the elements. Furthermore, the atoms of one algebra must correspond to the atoms in the isomorphic algebra.

If we wish to find an isomorphism between any boolean algebra, K, and an algebra of sets, the atoms of K must correspond to the singleton elements of the algebra of sets. This suggests that we try to define an isomorphism from K to the algebra P A of subsets of the set A of atoms of K. The following theorem shows that if K is finite, we can set up such an isomorphism.

Representation Theorem for Finite Boolean Algebras. We already have a natural correspondence between the atoms of K and the atoms of P A. We use the disjunctive normal form Theorem 2. Define the function f: The uniqueness of the normal form implies that each element of K has a unique image in P A and that f is a bijection. We still have to show that f is a morphism of boolean algebras. This proves that f is a boolean algebra isomorphism.

This follows from Theorem 2. The atoms of this algebra are the prime divisors 2, 5, and Do the divisors of 12 form a boolean algebra under gcd and lcm? Since the number of elements is not a power of 2, it cannot form a boolean algebra. Do the divisors of 24 form a boolean algebra under gcd and lcm? However, the poset diagram in Figure 2.

Hence, by Corollary 2. Poset diagram of the divisors of Prove the remaining parts of Proposition 2. Prove Theorem 2. That is, if X is a finite set with n elements, prove that P X contains 2n elements. Prove or give a counterexample to the following statements.

This shows that P X always contains more elements than X. What percentage, at least must have lost all four? One hundred students were questioned about their study habits. Seventy said they sometimes studied during the day, 55 said they sometimes studied during the night, and 45 said they sometimes studied during the weekend. How many did not study at all? If the zero element is the same as the unit element in a boolean algebra, prove that the algebra has only one element. Is this algebra isomorphic to the algebra of subsets of some set?

Draw the poset diagram for F1 , the boolean algebra of switching functions of one variable. Write down the truth tables for the following propositions. Which of these propositions are equivalent? Which of the following are tautologies, and which are contradictions? Harry broke the window if and only if he ran away and John was lying. John said that either Harry broke the window or Harry did not run away.

If Harry ran away, then he did not break the window. What conclusions can you come to? Draw circuits to realize the expressions in Exercises 2. Simplify the following expression and then draw a circuit for it. By looking at all the possible paths through the bridge circuit in Figure 2. Find a series-parallel circuit that is equivalent to the bridge circuit in Figure 2.

A hall light is controlled by two switches, one upstairs and one downstairs. Design a circuit so that the light can be switched on or off from the upstairs or the downstairs. A large room has three separate entrances, and there is a light switch by each entrance. Design a circuit that will allow the lights to be turned on or off by throwing any one switch. Design and simplify such a machine. Design and simplify a voting machine for five people.

Design a circuit for a light that is controlled by two independent switches A and B and a master switch C. C must always be able to turn the light on. When C is off, the light should be able to be turned on and off using A or B. A committee consists of a chairman A, and three other members, B, C, and D. If B, C, and D, are not unanimous in their voting, the chairman decides the vote. Design a voting machine for this committee and simplify it as much as possible.

Verify that the Venn diagram in Figure 2. Then use the diagram to simplify the circuit in Figure 2. Design four series-parallel circuits to multiply two numbers in binary form that have at most two digits each. Design a circuit that will turn an orange light on if exactly one of the four switches A, B, C, and D is on and a green light when all four are on. Five switches are set to correspond to a number in binary form that has at most five digits.

Design and simplify a circuit that will switch a light on if and only if the binary number is a perfect square. These relays are used to control the valves that add cold water, that let water out, and that heat the water in the pool.

Design and simplify a circuit that will perform the following tasks. If the temperature is correct but the level too high, it is to let water out.

If the temperature is correct but the level too low, it is to let in cold water and heat the water. If the pool is too warm, add cold water and, if the level is also too high, let water out at the same time.

If the pool is too cold but the level correct, heat the water; if the level is too low, heat the water and add cold water, and, if the level is too high, just let out the water. In a dual fashion to the disjunctive normal form, every boolean expression in n-variables can be written in its conjunctive normal form. Draw poset diagrams for the sets given in Exercises 2. D54 2.

D42 2. Describe the atoms in the boolean algebra Dn , gcd, lcm,. Suppose that A and B are elements of a boolean algebra. This is the kind of algebraic model that would be required to deal with transistor switching gates under transient conditions.

Analyze the effect of the circuit in Figure 2. One of the basic types of components of a digital computer is a flip-flop. One important use of a flip-flop is to store a binary digit.

An RS flip-flop is a circuit with two inputs, R and S, and one output, Q, corresponding to the state of the flip-flop. If both R and S are 0, the next state is the same as the previous state Q. It is assumed that R and S cannot both be 1 simultaneously. Verify that the NOR circuit in Figure 2. Q RS flip-flop. A JK flip-flop is similar to an RS flip-flop except that both inputs are allowed to be 1 simultaneously, and in this case the state Q changes to its opposite state.

In Chapter 5, we use group theory to determine all the symmetries that can occur in two- or three-dimensional space. This can be used, for example, to classify all the forms that chemical crystals can take.

If we have a large class of objects, some of which are equivalent under permutations or symmetries, we show, in Chapter 6, how groups can be used to count the nonequivalent objects. For example, we count the number of different switching functions of n variables if we allow permutations of the inputs. Historically, the basic ideas of group theory arose with the investigation of permutations of finite sets in the theory of equations.

One of the aims of mathematicians at the beginning of the nineteenth century was to find methods for solving polynomial equations of degree 5 and higher. Algorithms, involving the elementary arithmetical operations and the extraction of roots, were already known for solving all polynomial equations of degree less than 5; the formulas for solving quadratic equations had been known since Babylonian times, and cubic and quartic equations had been solved by various Italian mathematicians in the sixteenth century.

However, in , using the rudiments of group theory, the Norwegian Niels Abel — showed that some equations of the fifth degree could not be solved by any such algorithm. This theory, now called Galois theory, is beyond the scope of this book, but interested students should look at Stewart [35] after reading Chapter It was not until the s that the abstract definition of a group that we use today began to emerge.

It was soon discovered that this concept of a group was so universal that it cropped up in many different branches of mathematics and science.

The closure axiom is already implied by the definition of a binary operation; however, it is included because it is often overlooked otherwise. The product of any two of these elements is an element of G; thus G is closed under the operation. Multiplication is associative and commutative in G because multiplication of complex numbers is always associative and commutative.

The multiplication of any two elements of G can be represented by Table 3. The identity is 0, and the inverse of each element is its negative. The group axioms follow from Proposition 2.

TABLE 3. A group with only this one element is called trivial. Many important groups consist of functions. Given functions f: Composition is associative; that is, if h: Moreover, this operation has an identity: The identity function 1X: This inverse is unique when it exists.

When it exists see Theorem 3. Example 3. A translation of the plane R2 in the direction of the vector a, b is a function f: The identity is the identity transformation 1R2: In other words, an injective function never takes two different points to the same point.

A bijective function or one-to-one correspondence is a function that is both injective and surjective. A permutation or symmetry of a set X is a bijection from X to itself. Lemma 3. Theorem 3. Inversion Theorem. Suppose that h: Therefore, f is bijective. Conversely, suppose that f is bijective. We define the function h: Since f is injective, there is only one such element x.

This group is called the symmetric group or permutation group of X. It follows from Lemma 3. The composition of functions is always associative, and the identity of S X is the identity function 1X: The inversion theorem Theorem 3.

The use of the term symmetry to describe the bijection f agrees with one of our everyday uses of the word. Since the composition of functions is not generally commutative, S X is not usually an abelian group. A nonsingular linear transformation of the plane is a bijective function of the form f: It can be verified that the composition of two such linear transformations is again of the same type.

Besides talking about the symmetries of a distinct set of elements, we often refer, in everyday language, to a geometric object or figure as being symmetrical. We now make this notion more mathematically precise. If F is a figure in the plane or in space, a symmetry of the figure F or isometry of F is a bijection f: One can visualize this operation by imagining F to be a solid object that can be picked up and turned in some manner so that it assumes a configuration identical to the one it had in its original position.

For example, the design on the left of Figure 3. The design in the center of Figure 3. Symmetrical designs. However, both the one-third rotation and interchanging two vertices are symmetries of the equilateral triangle on the right in Figure 3. The rotation can be performed as a physical motion within the plane of the triangle and so is called a proper symmetry or a proper rotation , while the reflection can only be accomplished as a physical motion by moving the triangle outside its plane an improper symmetry or an improper rotation.

The set of all symmetries of a geometric figure forms a group under composition because the composition and inverse of two distance-preserving functions is distance preserving.

Write down the table for the group of symmetries of a rectangle with unequal sides. Label the corners of the rectangle 1, 2, 3, and 4 as in Figure 3. Any symmetry of the rectangle will send corner points to corner points and so will permute the corners among themselves.

This symmetry can also be considered as a rotation of the rectangle through half a revolution about this horizontal axis. There is a similar symmetry, b, about the vertical axis through the center. A third proper symmetry, c, is obtained by rotating the rectangle in its plane through half a revolution about its center. Finally, the identity map, e, is a symmetry. These are the only symmetries because it can be verified that any other bijection between the corners will not preserve distances.

The other products can be calculated similarly. We have seen that the group operation can be denoted by various symbols, the most common being multiplication, composition, and addition. It is conventional b 1 2 a c 4 Figure 3. Whenever the operation in a group is clearly understood, we denote the group just by its underlying set.

This should cause no confusion because Z, Q, R, and C are not groups under multiplication since the element 0 has no multiplicative inverse. The symmetric group of X is denoted just by S X , the operation of composition being understood. Moreover, if we refer to a group G without explicitly defining the group or the operation, it can be assumed that the operation in G is multiplication.

We now prove two propositions that will enable us to manipulate the elements of a group more easily. Recall from Proposition 2. We first show that the inverse of any element of a group is unique. Proposition 3. Then, if an element a has an inverse, this inverse is unique. But a is such an element, and by Proposition 3. A more everyday example is the operation of putting on socks and shoes.

To reverse the procedure, the shoes are taken off first and then the socks. Finally, we set a 0 to be the identity, again as for numbers. Such a group is called a subgroup. Since H is nonempty, it contains at least one element, say h. However, when H is finite, the following result shows that it is sufficient just to check condition i.

In the group of translations of the plane Example 3. Remember that addition is the operation in all these groups. This example shows that Proposition 3. The relation of being a subgroup is transitive. In fact, for any group G, the inclusion relation between the subgroups of G is a partial order relation.

Draw the poset diagram of the subgroups of the group of symmetries of a rectangle. By looking at the table of this group in Table 3.

Subgroups of the group of symmetries of a rectangle. The poset diagram of subgroups is given in Figure 3. G is called a finite group if G is finite, and G is called an infinite group otherwise. An important class of groups consists of those for which every element can be written as a power positive or negative of some fixed element.

The element g is called a generator of the cyclic group. If no such r exists, the order of the element is said to be infinite. Note the difference between the order of an element and the order of a group. The group has order 4. The group has infinite order. The next two results show how the division algorithm for integers see Appendix 2 is used in group theory. Let a be an element of order r in a group G. In this case, H is infinite. If the finite group G is of order n and has an element g of order n, then G is a cyclic group generated by g.

From the previous proposition we know that H , the subgroup of G generated by g, has order n. Therefore, H is a subset of the finite set G with the same number of elements. Show that the Klein 4-group of symmetries of a rectangle, described in Example 3.

In the Klein 4-group, the identity has order 1, whereas all the other elements have order 2. As it has no element of order 4, it cannot be cyclic. We therefore say that this group can be generated by the two elements a and b. This group is denoted by Cn. This is the group of those symmetries of the regular n-gon that can be performed in the plane, that is, without turning the n-gon over. Label the vertices 1 through n as in Figure 3. Under any symmetry, the center must be fixed, and the vertex 1 can be taken to any of the n vertices.

The image of 1 determines the rotation; hence the group is of order n. Then g has order n, and by Theorem 3. We call this group the dihedral group and denote it by Dn. Show that the dihedral group, Dn , is of order 2n and is not cyclic. Label the vertices 1 to n in a counterclockwise direction around the n-gon. The element g generates the group Cn , which is a cyclic subgroup of Dn.

Elements of Cn. Elements of Dn. The vertex 1 can be taken to any of the n vertices, and then 2 must be taken to one of the two vertices adjacent to the image of 1.

Hence Dn has order 2n. If the image of 2 is r, the symmetry is g r h. Figure 3. Therefore, this group cannot be cyclic. Draw the group table for C4 and D4. Since C4 is a subgroup of D4 , the table for C4 appears inside the dashed lines in the top left corner. Therefore, g r h always has order 2. For instance, in Example 3. This defines a function f: Since this function preserves the operations, it is a morphism of groups.

Two groups are isomorphic if their structures are essentially the same. We often use the notation f: Many authors use homomorphism instead of morphism but we prefer the simpler terminology.

A group isomorphism is a bijective group morphism. If G and H are any two groups, the trivial function that maps every element of G to the identity of H is always a morphism. Isomorphic groups share exactly the same properties, and we sometimes identify the groups via the isomorphism and give them the same name. Besides preserving the operations of a group, the following result shows that morphisms also preserve the identity and inverses.

Let f: Hence i follows by cancellation in H Proposition 3. Cyclic groups of the same order are isomorphic. Hence the function f: Then the function f: Any morphism, f: Corresponding elements under a group isomorphism have the same order. Suppose that g and h have orders m and n, respectively, where m is finite.

Is D2 isomorphic to C4 or the Klein 4-group of symmetries of a rectangle? Compare the orders of the elements given in Table 3. By Proposition 3. Table 3. The permutation groups of two sets, with the same number of elements, are isomorphic. There are n possible choices for the image of 1 under a bijection.

To calculate this, we start at 3 1 2 3 2 1 the right and trace the image of each element under the composition. In general, if a1 , a2 ,. An r-cycle in Sn has order r. We can either calculate a product of cycles from the permutation representation or we can use the cycle representation directly. In calculating such a composition, we begin at the right and work our way left. This is also the orbit of 3, 4, and 8. This orbit gives rise to the cycle 1 3 8 4. We can picture the orbits and their corresponding cycles as in Figure 3.

Since no number is in two different cycles, these cycles are called disjoint. The identity permutation is usually just written as 1.

Every permutation can be written as a product of disjoint cycles. Corollary 3. The order of a permutation is the least common multiple of the lengths of its disjoint cycles.

Show that D3 is isomorphic to S3 and write out the table for the latter group. D3 is the group of symmetries of an equilateral triangle, and any symmetry induces a permutation of the vertices. Hence f is a morphism.

We shade the underside of the triangle and mark the corner near vertex 1 to illustrate how the triangle moves. To visualize this, imagine a triangular jigsaw puzzle piece and consider all possible ways of fitting this piece into a triangular hole. Any proper rotation will leave the white side uppermost, whereas an improper rotation will leave the shaded side uppermost.

The six permutations are all distinct; thus f is a bijection and an isomorphism between D3 and S3. The group table for S3 is given in Table 3. We are going to determine an easy method for deciding which is the case.

A 2cycle is called a transposition, and surprisingly, much of the discussion centers around determining the parity of these transpositions. We are going to show that every transposition is odd. Let D denote the discriminant in n variables x1 , x2 ,. Then D factors as follows: Every transposition is odd.

The proof of the next result is a straightforward verification. This gives us the desired parity test. Parity Theorem. An n-cycle is an even permutation if n is odd and an odd permutation if n is even. It follows from Theorem 3. In fact, we have Theorem 3.

Hence n! We do this by finding a bijection f: Thus f is surjective, as required. Every even permutation can be written as a product of 3-cycles not necessarily disjoint. By Theorem 3. We show that any product of two transpositions is a product of 3-cycles. If these two transpositions are identical, their product is the identity. Arthur Cayley — was the first mathematician to deal with groups abstractly in terms of axioms, but he showed that any abstract group can be considered as a subgroup of a symmetric group.

Hence, in some sense, if you know all about symmetry groups and permutation groups, you know all about group theory. This is clearly surjective.

If G is a finite group of order n, then G is isomorphic to a subgroup of Sn. This follows because S G is isomorphic to Sn. Give reasons for your answers. For Exercises 3. These patterns are repeated indefinitely in both directions. Prove that the relation of being a subgroup is a partial order on the set of subgroups of a group G.

Draw the poset diagram of the subgroups of C6. Draw the poset diagram of the subgroups of S3. Give reasons. Find the orders of all the elements in A4. Show that G is a non-abelian group of order 8.

Is it isomorphic to D4 or the quaternion group Q? Prove that a group of even order always has at least one element of order 2. Find a subgroup of S7 of order Find a subgroup of S5 of order 3. Find a subgroup of A4 isomorphic to the Klein 4-group.

Multiply out the permutations in Exercises 3. Find the order of each permutation and state whether the permutation is even or odd. These will form subgroups of S4 , called the symmetry groups of the polynomials. Describe the group of proper rotations of the tetrahedron with vertices 0, 0, 0 , 1, 0, 0 , 0, 1, 0 , and 0, 0, 1 in R3. Prove that f is an onto f: What is the number of generators of the cyclic group Cn?

Can you generalize this result? If there were 50 cards, what would be the least number? Show that Z G is an abelian subgroup of G. Find the center of D3. Find the center of D4. Prove that Sn is generated by the elements 12 , 23 , 34 ,.

The bottom right corner square is removed, and the other squares are labeled as in Figure 3. By sliding the squares around without lifting them up , show that the set of possible permutations that can be obtained with the bottom right square blank is precisely A There is no known easy proof that all elements in A15 must occur. Which of the positions of the puzzle shown in Figure 3. An automorphism of a group G is an isomorphism from G to itself.

Prove that the set of all automorphisms of G forms a group under composition. Find the automorphism group of the Klein 4-group. Find the automorphism group of C3. Find the automorphism group of C4. Find the automorphism group of S3. Let F be the set of such words together with the empty word, which is denoted by 1.

The operation of concatenation places one word after another. F is called the free group on two generators. Is F abelian? One such technique is the construction of the quotient set of an algebraic object by means of an equivalence relation on the underlying set.

This quotient construction can be applied to numerous algebraic structures, including groups, boolean algebras, and vector spaces. In this chapter we introduce the concept of an equivalence relation and go on to apply this to groups. We study the implications of these two theorems and classify the groups of low order.

We say that a is related to b under R if the pair a, b belongs to the subset, and we write this as aRb. Any function f: However, relations are much more general than functions. One element can be related to many elements or to no elements at all. A relation E on a set S is called an equivalence relation if the following conditions hold.

Proposition 4. Then xEa and so xEb by transitivity. Hence S is the disjoint union of all the equivalence classes. The previous proposition shows that any equivalence relation partitions the set into its equivalence classes. Each element of the set belongs to one and only one equivalence class. It can also be shown that every partition of a set gives rise to an equivalence relation whose classes are precisely the subsets in the partition. Example 4. Let n be a fixed positive integer and a and b any two integers.

The set of equivalence classes is called the set of integers modulo n and is denoted by Zn. Hence congruence modulo n is an equivalence relation on Z. One set of equivalence classes that is introduced in elementary school is the set of rational numbers. Students soon become used to the fact that 12 and 36 represent the same rational number.

We need to use the concept of equivalence class to define a rational number precisely. We denote the equivalence class [ a, b ] by ab. Two series-parallel circuits involving the switches A1 , A2 ,. This is an equivalence relation, and the equivalence classes are the 22 distinct types of circuits controlled by n switches.

We now generalize this notion and define congruence in any group modulo one of its subgroups. We are interested in the equivalence classes, which we call cosets. The element a is called a representative of the coset Ha.

The relation is transitive. Find the right cosets of A3 in S3. Take any element not in the subgroup, say Since the right cosets form a partition of S3 and the two cosets above contain all the elements of S3 , it follows that these are the only two cosets. H itself is one coset. These two cosets have not exhausted all the elements of C12 , so pick an element, say g 2 , which is not in H or Hg.

We use this result to prove the famous theorem of Joseph Lagrange — There is a bijection between any two right cosets of H in G. Let Ha be a right coset of H in G. We produce a bijection between Ha and H , from which it follows that there is a bijection between any two right cosets. By Lemma 4. Therefore, H divides G. Corollary 4. If G is a finite group with subgroup H , then G: If a is an element of a finite group G, then the order of a divides the order of G.

If G is a group of prime order, then G is cyclic. By Corollary 4. But the only element of order 1 is the identity. Hence by Theorem 3. That is, if k is a divisor of the order of G, it does not necessarily follow that G has a subgroup of order k. A4 is a group of order 12 having no subgroup of order 6. If a subgroup contains a 3-cycle abc , it must also contain its inverse acb. If a subgroup of order 6 exists, it must contain the identity and a product of two transpositions, because the odd number of nonidentity elements cannot be made up of 3-cycles and their inverses.

A subgroup of order 6 must also contain at least two 3-cycles because A4 only contains four elements that are not 3-cycles. Hence A4 contains no subgroup of order 6. The following lemma, of interest in its own right, will be needed. Lemma 4. But and are relatively prime by Theorem 11, d d d d n Appendix 2, so divides k by the same theorem.

If G is a cyclic group of order n, and if k divides n, then G has exactly one subgroup H of order k. As the following example shows, the left coset of an element does not necessarily equal the right coset. However, the left and right cosets of K are not all the same.

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