BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230

Solutions

Chapter
Section
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230
Chapter 3.2, Problem 5E
Textbook Problem
1 views

In Example 3 of Section 3.1, find elements a and b

of S ( A )

such that ( a b ) 1 a 1 b 1 .

From Example 3 of section 3.1: A = { 1 , 2 , 3 } and S ( A )

is a set of bijective functions defined on A .

To determine

The elements a and b of S(A) such that (ab)1a1b1

Explanation of Solution

Given information:

A={1,2,3} and S(A) set of functions defined from A to A.

Formula used:

1) Composition of function: Let f and g be the functions defined from A to A, then fg:AA, is defined as fg(x)=f[g(x)]

2) Definition of a group.

Suppose the binary operation is defined for element of a set G. The set G is a group with respect to , provided the following conditions hold.

1. G is closed under . That is xG and yG imply that xy is in G.

2. is associative. For all x,y,z in G, x(yz)=(xy)z.

3. G has an identity element e. There is an e in G such that xe=ex=x for all xG.

4. G contains inverses. For each aG, there exists bG such that ab=ba=e.

Calculation:

Let A={1,2,3}, S(A) be a set of bijective functions defined on A.

f(1) have three choices, f(2) have two choices and f(3) has only one choice.

So there are 3!=321=6 different mappings given by,

e={e(1)=1e(2)=2e(3)=3, σ={σ(1)=2σ(2)=1σ(3)=3, ρ={ρ(1)=2ρ(2)=3ρ(3)=1

γ={γ(1)=3γ(2)=2γ(3)=1, τ={τ(1)=3τ(2)=1τ(3)=2, δ={δ(1)=1δ(2)=3δ(3)=2

Thus, S(A)={e,ρ,τ,σ,γ,δ}.

In constructing the table for S(A), list the elements of S(A) in a column at the left

and in a row at the top.

Let e as an identity map, therefore

eρ=ρ,eτ=τ,eσ=σ,eγ=γ,eδ=δ

Similarly,

ρe=ρ,τe=τ,σe=σ,γe=γ,δe=δ.

When the product ρ2=ρρ is computed, then

ρ2(1)=ρ[ρ(1)]=ρ(2)=3ρ2(2)=ρ[ρ(2)]=ρ(3)=1ρ2(3)=ρ[ρ(3)]=ρ(1)=2ρρ=τ

Similarly, ρσ=γ, σρ=δ and so on.

Therefore, Cayley’s table for S(A) is given by,

eρτσγδeeρτσγδρρτeγδσττeρδσγσσδγeτργγσδρeτδδγστρe

Composition of e with other elements of S(A) satisfy the property,

(ab)1=a1b1

From the Cayley’s table,

a.

(ρρ)1=τ1=ρ,ρ1ρ1=ττ=ρ(ρρ)1=ρ1ρ1

b.

(ρτ)1=e1=e,ρ1τ1=τρ=e(ρτ)1=ρ1τ1

c.

(ρσ)1=γ1=γ,ρ1σ1=τσ=δ(ρσ)1ρ1σ1

d.

(ργ)1=δ1=δ,ρ1γ1=τγ=σ(ργ)1ρ1γ1

e.

(ρδ)1=σ1=σ,ρ1δ1=τδ=γ(ρδ)1ρ1δ1

f.

(τρ)1=e1=e,τ1ρ1=ρτ=e(τρ)1=τ1ρ1

g.

(ττ)1=ρ1=τ,τ1τ1=ρρ=τ(ττ)1=τ1τ1

h.

(τσ)1=δ1=δ,τ1σ1=ρσ=γ(τσ)1τ1σ1

i

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Chapter 3 Solutions

Elements Of Modern Algebra
Show all chapter solutions
add
Ch. 3.1 - True or False Label each of the following...Ch. 3.1 - In Exercises 114, decide whether each of the given...Ch. 3.1 - Exercises In Exercises , decide whether each of...Ch. 3.1 - In Exercises 114, decide whether each of the given...Ch. 3.1 - Exercises In Exercises, decide whether each of...Ch. 3.1 - In Exercises 114, decide whether each of the given...Ch. 3.1 - Exercises In Exercises, decide whether each of...Ch. 3.1 - Exercises In Exercises, decide whether each of...Ch. 3.1 - Exercises In Exercises, decide whether each of...Ch. 3.1 - Exercises In Exercises, decide whether each of...Ch. 3.1 - In Exercises 114, decide whether each of the given...Ch. 3.1 - Exercises In Exercises, decide whether each of...Ch. 3.1 - Exercises In Exercises, decide whether each of...Ch. 3.1 - In Exercises 114, decide whether each of the given...Ch. 3.1 - In Exercises 114, decide whether each of the given...Ch. 3.1 - In Exercises and, the given table defines an...Ch. 3.1 - In Exercises 15 and 16, the given table defines an...Ch. 3.1 - In Exercises, let the binary operation be defined...Ch. 3.1 - In Exercises, let the binary operation be defined...Ch. 3.1 - In Exercises, let the binary operation be defined...Ch. 3.1 - In Exercises 1724, let the binary operation be...Ch. 3.1 - In Exercises 1724, let the binary operation be...Ch. 3.1 - In Exercises, let the binary operation be defined...Ch. 3.1 - In Exercises, let the binary operation be defined...Ch. 3.1 - In Exercises, let the binary operation be defined...Ch. 3.1 - In Exercises, decide whether each of the given...Ch. 3.1 - In Exercises, decide whether each of the given...Ch. 3.1 - In Exercises 2532, decide whether each of the...Ch. 3.1 - In Exercises 2532, decide whether each of the...Ch. 3.1 - In Exercises, decide whether each of the given...Ch. 3.1 - In Exercises, decide whether each of the given...Ch. 3.1 - In Exercises, decide whether each of the given...Ch. 3.1 - In Exercises, decide whether each of the given...Ch. 3.1 - a. Let G={ [ a ][ a ][ 0 ] }n. Show that G is a...Ch. 3.1 - 34. Let be the set of eight elements with...Ch. 3.1 - 35. A permutation matrix is a matrix that can be...Ch. 3.1 - Consider the matrices R=[ 0110 ] H=[ 1001 ] V=[...Ch. 3.1 - Prove or disprove that the set of all diagonal...Ch. 3.1 - 38. Let be the set of all matrices in that have...Ch. 3.1 - 39. Let be the set of all matrices in that have...Ch. 3.1 - 40. Prove or disprove that the set in Exercise ...Ch. 3.1 - 41. Prove or disprove that the set in Exercise ...Ch. 3.1 - 42. For an arbitrary set , the power set was...Ch. 3.1 - Write out the elements of P(A) for the set A={...Ch. 3.1 - Let A={ a,b,c }. Prove or disprove that P(A) is a...Ch. 3.1 - 45. Let . Prove or disprove that is a group with...Ch. 3.1 - In Example 3, the group S(A) is nonabelian where...Ch. 3.1 - 47. Find the additive inverse of in the given...Ch. 3.1 - Find the additive inverse of [ [ 2 ][ 3 ][ 4 ][ 1...Ch. 3.1 - 49. Find the multiplicative inverse of in the...Ch. 3.1 - 50. Find the multiplicative inverse of in the...Ch. 3.1 - Prove that the Cartesian product 24 is an abelian...Ch. 3.1 - Let G1 and G2 be groups with respect to addition....Ch. 3.2 - True or False Label each of the following...Ch. 3.2 - True or False Label each of the following...Ch. 3.2 - Label each of the following statements as either...Ch. 3.2 - True or False Label each of the following...Ch. 3.2 - Label each of the following statements as either...Ch. 3.2 - Label each of the following statements as either...Ch. 3.2 - 1.Prove part of Theorem . Theorem 3.4: Properties...Ch. 3.2 - Prove part c of Theorem 3.4. Theorem 3.4:...Ch. 3.2 - Prove part e of Theorem 3.4. Theorem 3.4:...Ch. 3.2 - An element x in a multiplicative group G is called...Ch. 3.2 - 5. In Example 3 of Section 3.1, find elements and...Ch. 3.2 - 6. In Example 3 of section 3.1, find elements and ...Ch. 3.2 - 7. In Example 3 of Section 3.1, find elements and...Ch. 3.2 - In Example 3 of Section 3.1, find all elements a...Ch. 3.2 - 9. Find all elements in each of the following...Ch. 3.2 - 10. Prove that in Theorem , the solutions to the...Ch. 3.2 - Let G be a group. Prove that the relation R on G,...Ch. 3.2 - Suppose that G is a finite group. Prove that each...Ch. 3.2 - In Exercises and , part of the multiplication...Ch. 3.2 - In Exercises 13 and 14, part of the multiplication...Ch. 3.2 - 15. Prove that if for all in the group , then ...Ch. 3.2 - Suppose ab=ca implies b=c for all elements a,b,...Ch. 3.2 - 17. Let and be elements of a group. Prove that...Ch. 3.2 - Let a and b be elements of a group G. Prove that G...Ch. 3.2 - Use mathematical induction to prove that if a is...Ch. 3.2 - 20. Let and be elements of a group . Use...Ch. 3.2 - Let a,b,c, and d be elements of a group G. Find an...Ch. 3.2 - Use mathematical induction to prove that if...Ch. 3.2 - 23. Let be a group that has even order. Prove that...Ch. 3.2 - 24. Prove or disprove that every group of order is...Ch. 3.2 - 25. Prove or disprove that every group of order is...Ch. 3.2 - 26. Suppose is a finite set with distinct...Ch. 3.2 - 27. Suppose that is a nonempty set that is closed...Ch. 3.2 - Reword Definition 3.6 for a group with respect to...Ch. 3.2 - 29. State and prove Theorem for an additive...Ch. 3.2 - 30. Prove statement of Theorem : for all integers...Ch. 3.2 - 31. Prove statement of Theorem : for all integers...Ch. 3.2 - Prove statement d of Theorem 3.9: If G is abelian,...Ch. 3.3 - Label each of the following statements as either...Ch. 3.3 - True or false Label each of the following...Ch. 3.3 - True or false Label each of the following...Ch. 3.3 - True or false Label each of the following...Ch. 3.3 - True or false Label each of the following...Ch. 3.3 - True or false Label each of the following...Ch. 3.3 - True or false Label each of the following...Ch. 3.3 - True or false Label each of the following...Ch. 3.3 - True or false Label each of the following...Ch. 3.3 - True or false Label each of the following...Ch. 3.3 - True or false Label each of the following...Ch. 3.3 - Let S(A)={ e,,2,,, } be as in Example 3 in section...Ch. 3.3 - Decide whether each of the following sets is a...Ch. 3.3 - 3. Consider the group under addition. List all...Ch. 3.3 - 4. List all the elements of the subgroupin the...Ch. 3.3 - 5. Exercise of section shows that is a group...Ch. 3.3 - 6. Let be , the general linear group of order...Ch. 3.3 - 7. Let be the group under addition. List the...Ch. 3.3 - Find a subset of Z that is closed under addition...Ch. 3.3 - 9. Let be a group of all nonzero real numbers...Ch. 3.3 - 10. Let be an integer, and let be a fixed...Ch. 3.3 - 11. Let be a subgroup of, let be a fixed element...Ch. 3.3 - Prove or disprove that H={ hGh1=h } is a subgroup...Ch. 3.3 - 13. Let be an abelian group with respect to...Ch. 3.3 - Prove that each of the following subsets H of...Ch. 3.3 - 15. Prove that each of the following subsets of ...Ch. 3.3 - Prove that each of the following subsets H of...Ch. 3.3 - 17. Consider the set of matrices, where ...Ch. 3.3 - Prove that SL(2,R)={ [ abcd ]|adbc=1 } is a...Ch. 3.3 - 19. Prove that each of the following subsets of ...Ch. 3.3 - For each of the following matrices A in SL(2,R),...Ch. 3.3 - 21. Let Be the special linear group of order ...Ch. 3.3 - 22. Find the center for each of the following...Ch. 3.3 - 23. Let be the equivalence relation on defined...Ch. 3.3 - 24. Let be a group and its center. Prove or...Ch. 3.3 - Let G be a group and Z(G) its center. Prove or...Ch. 3.3 - Let A be a given nonempty set. As noted in Example...Ch. 3.3 - (See Exercise 26) Let A be an infinite set, and...Ch. 3.3 - 28. For each, define by for. a. Show that is an...Ch. 3.3 - Let G be an abelian group. For a fixed positive...Ch. 3.3 - For fixed integers a and b, let S={ ax+byxandy }....Ch. 3.3 - 31. a. Prove Theorem : The center of a group is...Ch. 3.3 - Find the centralizer for each element a in each of...Ch. 3.3 - Prove that Ca=Ca1, where Ca is the centralizer of...Ch. 3.3 - 34. Suppose that and are subgroups of the group...Ch. 3.3 - 35. For an arbitrary in , the cyclic subgroup of...Ch. 3.3 - 36. Let , be an arbitrary nonempty collection of...Ch. 3.3 - 37. If is a group, prove that ,where is the...Ch. 3.3 - Find subgroups H and K of the group S(A) in...Ch. 3.3 - 39. Assume that and are subgroups of the abelian...Ch. 3.3 - 40. Find subgroups and of the group in example ...Ch. 3.3 - 41. Let be a cyclic group, . Prove that is...Ch. 3.3 - Reword Definition 3.17 for an additive group G....Ch. 3.3 - 43. Suppose that is a nonempty subset of a group ....Ch. 3.3 - 44. Let be a subgroup of a group .For, define the...Ch. 3.3 - Assume that G is a finite group, and let H be a...Ch. 3.4 - Label each of the following statements as either...Ch. 3.4 - Label each of the following statements as either...Ch. 3.4 - True or False Label each of the following...Ch. 3.4 - True or False Label each of the following...Ch. 3.4 - Label each of the following statements as either...Ch. 3.4 - Label each of the following statements as either...Ch. 3.4 - True or False Label each of the following...Ch. 3.4 - True or False Label each of the following...Ch. 3.4 - Label each of the following statements as either...Ch. 3.4 - True or False Label each of the following...Ch. 3.4 - Exercises 1. List all cyclic subgroups of the...Ch. 3.4 - Let G=1,i,j,k be the quaternion group. List all...Ch. 3.4 - Exercises 3. Find the order of each element of the...Ch. 3.4 - Find the order of each element of the group G in...Ch. 3.4 - The elements of the multiplicative group G of 33...Ch. 3.4 - Exercises 6. In the multiplicative group, find the...Ch. 3.4 - Exercises 7. Let be an element of order in a...Ch. 3.4 - Exercises 8. Let be an element of order in a...Ch. 3.4 - Exercises 9. For each of the following values of,...Ch. 3.4 - Exercises 10. For each of the following values of,...Ch. 3.4 - Exercises 11. According to Exercise of section,...Ch. 3.4 - For each of the following values of n, find all...Ch. 3.4 - Exercises 13. For each of the following values of,...Ch. 3.4 - Exercises 14. Prove that the set is cyclic...Ch. 3.4 - Exercises 15. a. Use trigonometric identities and...Ch. 3.4 - For an integer n1, let G=Un, the group of units in...Ch. 3.4 - let Un be the group of units as described in...Ch. 3.4 - Exercises 18. Let be the group of units as...Ch. 3.4 - Exercises 19. Which of the groups in Exercise are...Ch. 3.4 - Consider the group U9 of all units in 9. Given...Ch. 3.4 - Exercises 21. Suppose is a cyclic group of order....Ch. 3.4 - Exercises 22. List all the distinct subgroups of...Ch. 3.4 - Let G= a be a cyclic group of order 24. List all...Ch. 3.4 - Let G= a be a cyclic group of order 35. List all...Ch. 3.4 - Describe all subgroups of the group under...Ch. 3.4 - Find all generators of an infinite cyclic group G=...Ch. 3.4 - Exercises 27. Prove or disprove that each of the...Ch. 3.4 - Exercises 28. Let and be elements of the group....Ch. 3.4 - Let a and b be elements of a finite group G. Prove...Ch. 3.4 - Let G be a group and define the relation R on G by...Ch. 3.4 - Exercises 31. Let be a group with its...Ch. 3.4 - If a is an element of order m in a group G and...Ch. 3.4 - If G is a cyclic group, prove that the equation...Ch. 3.4 - Exercises 34. Let be a finite cyclic group of...Ch. 3.4 - Exercises 35. If is a cyclic group of order and ...Ch. 3.4 - Suppose that a and b are elements of finite order...Ch. 3.4 - Suppose that a is an element of order m in a group...Ch. 3.4 - Exercises 38. Assume that is a cyclic group of...Ch. 3.4 - Suppose a is an element of order mn in a group G,...Ch. 3.4 - Exercises 40. Prove or disprove: If every...Ch. 3.4 - Let G be an abelian group. Prove that the set of...Ch. 3.4 - Let d be a positive integer and (d) the Euler...Ch. 3.5 - Label each of the following statements as either...Ch. 3.5 - True or False Label each of the following...Ch. 3.5 - Label each of the following statements as either...Ch. 3.5 - True or False Label each of the following...Ch. 3.5 - Label each of the following statements as either...Ch. 3.5 - True or False Label each of the following...Ch. 3.5 - Label each of the following statements as either...Ch. 3.5 - True or False Label each of the following...Ch. 3.5 - Prove that if is an isomorphism from the group G...Ch. 3.5 - Let G1, G2, and G3 be groups. Prove that if 1 is...Ch. 3.5 - Exercises 3. Find an isomorphism from the additive...Ch. 3.5 - Let G=1,i,1,i under multiplication, and let G=4=[...Ch. 3.5 - Let H be the group given in Exercise 17 of Section...Ch. 3.5 - Exercises 6. Find an isomorphism from the additive...Ch. 3.5 - Find an isomorphism from the additive group to...Ch. 3.5 - Exercises 8. Find an isomorphism from the group ...Ch. 3.5 - Exercises 9. Find an isomorphism from the...Ch. 3.5 - Exercises 10. Find an isomorphism from the...Ch. 3.5 - The following set of matrices [ 1001 ], [ 1001 ],...Ch. 3.5 - Exercises 12. Prove that the additive group of...Ch. 3.5 - Consider the groups given in Exercise 12. Find an...Ch. 3.5 - Consider the additive group of real numbers....Ch. 3.5 - Consider the additive group of real numbers....Ch. 3.5 - Exercises 16. Assume that the nonzero complex...Ch. 3.5 - Let G be a group. Prove that G is abelian if and...Ch. 3.5 - Exercises 18. Suppose and let be defined by ....Ch. 3.5 - According to Exercise of Section, If n is a prime,...Ch. 3.5 - For each a in the group G, define a mapping ta:GG...Ch. 3.5 - For a fixed group G, prove that the set of all...Ch. 3.5 - Exercises 22. Let be a finite cyclic group of...Ch. 3.5 - Exercises 23. Assume is a (not necessarily...Ch. 3.5 - Let G be as in Exercise 23. Suppose also that ar...Ch. 3.5 - Exercises 25. Let be the multiplicative group of...Ch. 3.5 - Exercises 26. Use the results of Exercises and ...Ch. 3.5 - Exercises 27. Consider the additive groups , , and...Ch. 3.5 - Exercises 28. Let , , , and be groups with...Ch. 3.5 - Prove that any cyclic group of finite order n is...Ch. 3.5 - Exercises 30. For an arbitrary positive integer,...Ch. 3.5 - Prove that any infinite cyclic group is isomorphic...Ch. 3.5 - Let H be the group 6 under addition. Find all...Ch. 3.5 - Suppose that G and H are isomorphic groups. Prove...Ch. 3.5 - Exercises 34. Prove that if and are two groups...Ch. 3.5 - Exercises 35. Prove that any two groups of order ...Ch. 3.5 - Exercises 36. Exhibit two groups of the same...Ch. 3.5 - Let be an isomorphism from group G to group H....Ch. 3.5 - Exercises 38. If and are groups and is an...Ch. 3.5 - Suppose that is an isomorphism from the group G...Ch. 3.6 - Label each of the following statements as either...Ch. 3.6 - True or False Label each of the following...Ch. 3.6 - Label each of the following statements as either...Ch. 3.6 - Label each of the following statements as either...Ch. 3.6 - True or False Label each of the following...Ch. 3.6 - True or False Label each of the following...Ch. 3.6 - Label each of the following statements as either...Ch. 3.6 - True or False Label each of the following...Ch. 3.6 - Label each of the following statements as either...Ch. 3.6 - True or False Label each of the following...Ch. 3.6 - Each of the following rules determines a mapping...Ch. 3.6 - Each of the following rules determines a mapping ...Ch. 3.6 - 3. Consider the additive groups of real numbers...Ch. 3.6 - Consider the additive group and the...Ch. 3.6 - 5. Consider the additive group and define...Ch. 3.6 - Consider the additive groups 12 and 6 and define...Ch. 3.6 - Consider the additive groups 8 and 4 and define...Ch. 3.6 - 8. Consider the additive groups and . Define by...Ch. 3.6 - 9. Let be the additive group of matrices over...Ch. 3.6 - Rework exercise 9 with G=GL(2,), the general...Ch. 3.6 - 11. Let be , and let be the group of nonzero real...Ch. 3.6 - Consider the additive group of real numbers. Let ...Ch. 3.6 - Find an example of G, G and such that G is a...Ch. 3.6 - 14. Let be a homomorphism from the group to the...Ch. 3.6 - 15. Prove that on a given collection of groups,...Ch. 3.6 - 16. Suppose that and are groups. If is a...Ch. 3.6 - 17. Find two groups and such that is a...Ch. 3.6 - Suppose that is an epimorphism from the group G...Ch. 3.6 - 19. Let be a homomorphism from a group to a group...Ch. 3.6 - 20. If is an abelian group and the group is a...Ch. 3.6 - 21. Let be a fixed element of the multiplicative...Ch. 3.6 - 22. With as in Exercise , show that , and describe...Ch. 3.6 - Assume that is a homomorphism from the group G to...Ch. 3.6 - 24. Assume that the group is a homomorphic image...Ch. 3.6 - Let be a homomorphism from the group G to the...

Additional Math Textbook Solutions

Find more solutions based on key concepts
Show solutions add
Add: (8)+(2)

Elementary Technical Mathematics

Mobile DEVICE Usage The average time U.S. adults spent per day on mobile devices (in minutes) for the years 200...

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

In Problems 1-8, find the derivative of each function. 8.

Mathematical Applications for the Management, Life, and Social Sciences

Finding a Limit In Exercises 5-18, find the limit. limx3(x2+3x)

Calculus: Early Transcendental Functions (MindTap Course List)

For a given M 0, the corresponding in the definition of limx012x4= is: a) 2M4 b) 2M4 c) 12M4 d) 12M4

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th