Elements Of Modern Algebra
8th Edition
ISBN: 9781285463230
Author: Gilbert, Linda, Jimmie
Publisher: Cengage Learning,
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Chapter 3.4, Problem 42E
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Elements Of Modern Algebra
Ch. 3.1 - True or False
Label each of the following...Ch. 3.1 - True or False
Label each of the following...Ch. 3.1 - Label each of the following statements as either...Ch. 3.1 - True or False
Label each of the following...Ch. 3.1 - Prob. 5TFECh. 3.1 - True or False
Label each of the following...Ch. 3.1 - Label each of the following statements as either...Ch. 3.1 - Prob. 8TFECh. 3.1 - True or False
Label each of the following...Ch. 3.1 - True or False
Label each of the following...
Ch. 3.1 - True or False
Label each of the following...Ch. 3.1 - Prob. 1ECh. 3.1 - Prob. 2ECh. 3.1 - Prob. 3ECh. 3.1 - Prob. 4ECh. 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 - Prob. 11ECh. 3.1 - Prob. 12ECh. 3.1 - Prob. 13ECh. 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 - Prob. 25ECh. 3.1 - Prob. 26ECh. 3.1 - Prob. 27ECh. 3.1 - Prob. 28ECh. 3.1 - Prob. 29ECh. 3.1 - Prob. 30ECh. 3.1 - Prob. 31ECh. 3.1 - Prob. 32ECh. 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 - Prob. 48ECh. 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 - Prob. 7TFECh. 3.3 - Prob. 8TFECh. 3.3 - Prob. 9TFECh. 3.3 - Prob. 10TFECh. 3.3 - Prob. 11TFECh. 3.3 - Prob. 1ECh. 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 - Prob. 35ECh. 3.3 - Prob. 36ECh. 3.3 - Prob. 37ECh. 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 - Prob. 18ECh. 3.4 - Prob. 19ECh. 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 - Prob. 26ECh. 3.4 - Prob. 27ECh. 3.4 - Prob. 28ECh. 3.4 - Let a and b be elements of a finite group G. Prove...Ch. 3.4 - Prob. 30ECh. 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 - Prob. 34ECh. 3.4 - Prob. 35ECh. 3.4 - Prob. 36ECh. 3.4 - Prob. 37ECh. 3.4 - Exercises
38. Assume that is a cyclic group of...Ch. 3.4 - Prob. 39ECh. 3.4 - Prob. 40ECh. 3.4 - Let G be an abelian group. Prove that the set of...Ch. 3.4 - Prob. 42ECh. 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 - Prob. 8TFECh. 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 - Prob. 5ECh. 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 - Prob. 17ECh. 3.5 - Exercises
18. Suppose and let be defined by ....Ch. 3.5 - Prob. 19ECh. 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 - Prob. 24ECh. 3.5 - Prob. 25ECh. 3.5 - Prob. 26ECh. 3.5 - Exercises
27. Consider the additive groups , , and...Ch. 3.5 - Prob. 28ECh. 3.5 - Prob. 29ECh. 3.5 - Exercises
30. For an arbitrary positive integer,...Ch. 3.5 - Prob. 31ECh. 3.5 - Prob. 32ECh. 3.5 - Suppose that G and H are isomorphic groups. Prove...Ch. 3.5 - Prob. 34ECh. 3.5 - Exercises
35. Prove that any two groups of order ...Ch. 3.5 - Prob. 36ECh. 3.5 - Prob. 37ECh. 3.5 - Prob. 38ECh. 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 - Prob. 13ECh. 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...
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- Find the order of each of the following elements in the multiplicative group of units . for for for forarrow_forwardIn the expansion of (5x+3y)n , each term has the form (nk)ankbk ,where k successively takes on the value 0,1,2....,n. If (nk)=(72) what is the corresponding term?arrow_forwardExercises 10. For each of the following values of, find all subgroups of the cyclic group under addition and state their order. a. b. c. d. e. f.arrow_forward
- True or False Label each of the following statements as either true or false. 6. Every permutation can be expressed as a product of disjoint cycles.arrow_forwardWrite 20 as the direct sum of two of its nontrivial subgroups.arrow_forwardLet G1 and G2 be groups with respect to addition. Define equality and addition in the Cartesian product by G1G2 (a,b)=(a,b) if and only if a=a and b=a (a,b)+(c,d)=(ac,bd) Where indicates the addition in G1 and indicates the addition in G2. Prove that G1G2 is a group with respect to addition. Prove that G1G2 is abelian if both G1 and G2 are abelian. For notational simplicity, write (a,b)+(c,d)=(a+c,b+d) As long as it is understood that the additions in G1 and G2 may not be the same binary operations. (Sec. 3.4,27, Sec. 3.5,14,15,27,28, Sec. 3.6,12, Sec. 5.1,51) Sec. 3.4,27 Prove or disprove that each of the following groups with addition as defined in Exercises 52 of section 3.1 is cyclic. a. 23 b. 24 Sec. 3.5,14,15,27,28, Consider the additive group of real numbers. Prove or disprove that each of the following mappings : is an automorphism. Equality and addition are defined on in Exercise 52 of section 3.1. a. (x,y)=(y,x) b. (x,y)=(x,y) Consider the additive group of real numbers. Prove or disprove that each of the following mappings : is an isomorphism. a. (x,y)=x b. (x,y)=x+y Consider the additive groups 2, 3, and 6. Prove that 6 is isomorphic to 23. Let G1, G2, H1, and H2 be groups with respect to addition. If G1 is isomorphic to H1 and G2 is isomorphic to H2, prove that G1G2 is isomorphic to H1H2. Sec. 3.6,12 Consider the additive group of real numbers. Let be a mapping from to , where equality and addition are defined in Exercise 52 of Section 3.1. Prove or disprove that each of the following mappings is a homomorphism. If is a homomorphism, find ker , and decide whether is an epimorphism or a monomorphism. a. (x,y)=xy b. (x,y)=2x Sec. 5.1,51 Let R and S be arbitrary rings. In the Cartesian product RS of R and S, define (r,s)=(r,s) if and only if r=r and s=s (r1,s1)+(r2,s2)=(r1+r2,s1+s2), (r1,s1)(r2,s2)=(r1r2,s1s2). a. Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of R and S and is denoted by RS. b. Prove that RS is commutative if both R and S are commutative. c. Prove that RS has a unity element if both R and S have unity elements. d. Give an example of rings R and S such that RS does not have a unity element.arrow_forward
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