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. Every homomorphism is an...Problem 2TFE:
True or False
Label each of the following statements as either true or false.
Every isomorphism is a...Problem 3TFE:
Label each of the following statements as either true or false. Every endomorphism is an...Problem 4TFE:
Label each of the following statements as either true or false. Every epimorphism is an...Problem 5TFE:
True or False
Label each of the following statements as either true or false.
Every monomorphism is...Problem 6TFE:
True or False
Label each of the following statements as either true or false.
Every isomorphism is...Problem 7TFE:
Label each of the following statements as either true or false. The relation of being a homomorphic...Problem 8TFE:
True or False
Label each of the following statements as either true or false.
The kernel of a...Problem 9TFE:
Label each of the following statements as either true or false. It is possible to find at least one...Problem 10TFE:
True or False
Label each of the following statements as either true or false.
If there exists a...Problem 1E:
Each of the following rules determines a mapping :GG, where G is the group of all nonzero real...Problem 2E:
Each of the following rules determines a mapping from the additive group 4 to the additive group 2....Problem 3E:
3. Consider the additive groups of real numbers and complex numbers and define by . Prove that is...Problem 4E:
Consider the additive group and the multiplicative group G={ 1,i,1,i } and define :G by (n)=in....Problem 5E:
5. Consider the additive group and define by.
Prove that is a homomorphism and find ker .
Is an...Problem 6E:
Consider the additive groups 12 and 6 and define :126 by ([x]12)=[x]6. Prove that is a homomorphism...Problem 7E:
Consider the additive groups 8 and 4 and define :84 by ([x]8)=[x]4. Prove that is a homomorphism...Problem 8E:
8. Consider the additive groups and . Define by .
Prove that is a homomorphism and find ker ....Problem 9E:
9. Let be the additive group of matrices over and the additive group of real numbers. Define by
...Problem 10E:
Rework exercise 9 with G=GL(2,), the general linear group of order 2 over , and G= under addition....Problem 11E:
11. Let be , and let be the group of nonzero real numbers under multiplication. Prove that the...Problem 12E:
Consider the additive group of real numbers. Let be a mapping from to , where equality and...Problem 14E:
14. Let be a homomorphism from the group to the group .
Prove part a of Theorem : If denotes the...Problem 15E:
15. Prove that on a given collection of groups, the relation of being a homomorphic image has the...Problem 16E:
16. Suppose that and are groups. If is a homomorphic image of , and is a homomorphic image of ,...Problem 17E:
17. Find two groups and such that is a homomorphic image of but is not a homomorphic image of ....Problem 18E:
Suppose that is an epimorphism from the group G to the group G. Prove that is an isomorphism if...Problem 20E:
20. If is an abelian group and the group is a homomorphic image of , prove that is abelian.
Problem 21E:
21. Let be a fixed element of the multiplicative group . Definefrom the additive group to by for...Problem 22E:
22. With as in Exercise , show that , and describe the kernel of .
Exercise 21:
21. Let be a fixed...Problem 23E:
Assume that is a homomorphism from the group G to the group G. Prove that if H is any subgroup of...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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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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