ELEMENTS OF MODERN ALGEBRA8th EditionGilbert Publisher: CENGAGE LISBN: 9780357671139
ELEMENTS OF MODERN ALGEBRA
Solutions for ELEMENTS OF MODERN ALGEBRA
Problem 1TFE:
Label each of the following statements as either true or false. The order of the identity element in...Problem 2TFE:
Label each of the following statements as either true or false. Every cyclic group is abelian.Problem 3TFE:
True or False
Label each of the following statements as either true or false.
3. Every abelian group...Problem 4TFE:
True or False
Label each of the following statements as either true or false.
4. If a subgroup of a...Problem 5TFE:
Label each of the following statements as either true or false. Whether a group G is cyclic or not,...Problem 6TFE:
Label each of the following statements as either true or false. Every subgroup of a cyclic group is...Problem 7TFE:
True or False
Label each of the following statements as either true or false.
7. If there exists an...Problem 8TFE:
True or False
Label each of the following statements as either true or false.
8. Any group of order ...Problem 9TFE:
Label each of the following statements as either true or false. Any group of order 4 must be cyclic.Problem 10TFE:
True or False
Label each of the following statements as either true or false.
10. Let be an element...Problem 1E:
Exercises
1. List all cyclic subgroups of the group in Example of section.
Example 3.
We shall...Problem 3E:
Exercises
3. Find the order of each element of the group in Example of section.
Example 3.
We...Problem 4E:
Find the order of each element of the group G in Exercise 2. Let G=1,i,j,k be the quaternion group....Problem 5E:
The elements of the multiplicative group G of 33 permutation matrices are given in Exercise 35 of...Problem 7E:
Exercises
7. Let be an element of order in a group. Find the order of each of the following.
a. ...Problem 8E:
Exercises
8. Let be an element of order in a group. Find the order of each of the following.
a. ...Problem 9E:
Exercises
9. For each of the following values of, find all distinct generators of the cyclic group ...Problem 10E:
Exercises
10. For each of the following values of, find all subgroups of the cyclic group under...Problem 11E:
Exercises
11. According to Exercise of section, if is prime, the nonzero elements of form a...Problem 12E:
For each of the following values of n, find all distinct generators of the group Un described in...Problem 13E:
Exercises
13. For each of the following values of, find all subgroups of the group described in...Problem 14E:
Exercises
14. Prove that the set
is cyclic subgroup of the group . (Sec. )
7. Find an isomorphism ...Problem 15E:
Exercises
15. a. Use trigonometric identities and mathematical induction to prove that
for all...Problem 16E:
For an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have...Problem 17E:
let Un be the group of units as described in Exercise16. Prove that [ a ]Un if and only if a and n...Problem 20E:
Consider the group U9 of all units in 9. Given that U9 is a cyclic group under multiplication, find...Problem 21E:
Exercises
21. Suppose is a cyclic group of order. Determine the number of generators of for each...Problem 22E:
Exercises
22. List all the distinct subgroups of each group in Exercise.
Exercise
21. Suppose is...Problem 23E:
Let G= a be a cyclic group of order 24. List all elements having each of the following orders in G....Problem 24E:
Let G= a be a cyclic group of order 35. List all elements having each of the following orders in G....Problem 25E:
Describe all subgroups of the group under addition.Problem 29E:
Let a and b be elements of a finite group G. Prove that a and a1 have the same order. Prove that a...Problem 31E:
Exercises
31. Let be a group with its center:
.
Prove that if is the only element of order in ,...Problem 33E:
If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.Problem 38E:
Exercises
38. Assume that is a cyclic group of order. Prove that if divides , then has a subgroup...Browse All Chapters of This Textbook
Chapter 1.1 - SetsChapter 1.2 - MappingsChapter 1.3 - Properties Of Composite Mappings (optional)Chapter 1.4 - Binary OperationsChapter 1.5 - Permutations And InversesChapter 1.6 - MatricesChapter 1.7 - RelationsChapter 2.1 - Postulates For The Integers (optional)Chapter 2.2 - Mathematical InductionChapter 2.3 - Divisibility
Chapter 2.4 - Prime Factors And Greatest Common DivisorChapter 2.5 - Congruence Of IntegersChapter 2.6 - Congruence ClassesChapter 2.7 - Introduction To Coding Theory (optional)Chapter 2.8 - Introduction To Cryptography (optional)Chapter 3.1 - Definition Of A GroupChapter 3.2 - Properties Of Group ElementsChapter 3.3 - SubgroupsChapter 3.4 - Cyclic GroupsChapter 3.5 - IsomorphismsChapter 3.6 - HomomorphismsChapter 4.1 - Finite Permutation GroupsChapter 4.2 - Cayley’s TheoremChapter 4.3 - Permutation Groups In Science And Art (optional)Chapter 4.4 - Cosets Of A SubgroupChapter 4.5 - Normal SubgroupsChapter 4.6 - Quotient GroupsChapter 4.7 - Direct Sums (optional)Chapter 4.8 - Some Results On Finite Abelian Groups (optional)Chapter 5.1 - Definition Of A RingChapter 5.2 - Integral Domains And FieldsChapter 5.3 - The Field Of Quotients Of An Integral DomainChapter 5.4 - Ordered Integral DomainsChapter 6.1 - Ideals And Quotient RingsChapter 6.2 - Ring HomomorphismsChapter 6.3 - The Characteristic Of A RingChapter 6.4 - Maximal Ideals (optional)Chapter 7.1 - The Field Of Real NumbersChapter 7.2 - Complex Numbers And QuaternionsChapter 7.3 - De Moivre’s Theorem And Roots Of Complex NumbersChapter 8.1 - Polynomials Over A RingChapter 8.2 - Divisibility And Greatest Common DivisorChapter 8.3 - Factorization In F [x]Chapter 8.4 - Zeros Of A PolynomialChapter 8.5 - Solution Of Cubic And Quartic Equations By Formulas (optional)Chapter 8.6 - Algebraic Extensions Of A Field
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More Editions of This Book
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EBK ELEMENTS OF MODERN ALGEBRA
8th Edition
ISBN: 9780100475755
Elements Of Modern Algebra
8th Edition
ISBN: 9781285463230
EBK ELEMENTS OF MODERN ALGEBRA
8th Edition
ISBN: 8220100475757
Elements Of Modern Algebra
8th Edition
ISBN: 9781285965918
Elements of Modern Algebra
5th Edition
ISBN: 9780534373511
Elements of Modern Algebra
6th Edition
ISBN: 9780534402648
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