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, where H is subgroup of a group G....Problem 2TFE:
True or false
Label each of the following statements as either true or false, where is subgroup of...Problem 3TFE:
True or false Label each of the following statements as either true or false, where H is subgroup of...Problem 4TFE:
True or false
Label each of the following statements as either true or false, where is subgroup of...Problem 5TFE:
True or false
Label each of the following statements as either true or false, where is subgroup of...Problem 6TFE:
True or false
Label each of the following statements as either true or false, where is subgroup of...Problem 2E:
Decide whether each of the following sets is a subgroup of G={ 1,1,i,i } under multiplication. If a...Problem 3E:
3. Consider the group under addition. List all the elements of the subgroup, and state its order.
Problem 4E:
4. List all the elements of the subgroupin the group under addition, and state its order.
Problem 5E:
5. Exercise of section shows that is a group under multiplication.
a. List the elements of the...Problem 6E:
6. Let be , the general linear group of order over under multiplication. List the elements of the...Problem 7E:
7. Let be the group under addition. List the elements of the subgroup of for the given, and...Problem 8E:
Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.Problem 9E:
9. Let be a group of all nonzero real numbers under multiplication. Find a subset of that is...Problem 11E:
11. Let be a subgroup of, let be a fixed element of , and let be the set of all elements of the...Problem 13E:
13. Let be an abelian group with respect to multiplication. Prove that each of the following...Problem 14E:
Prove that each of the following subsets H of M2(Z) is subgroup of the group M2(Z) under addition....Problem 15E:
15. Prove that each of the following subsets of is subgroup of the group ,the general linear...Problem 16E:
Prove that each of the following subsets H of GL(2,C) is subgroup of the group GL(2,C), the general...Problem 17E:
17. Consider the set of matrices, where
, ,
, ...Problem 18E:
Prove that SL(2,R)={ [ abcd ]|adbc=1 } is a subgroup of GL(2,R), the general linear group of order 2...Problem 20E:
For each of the following matrices A in SL(2,R), list the elements of A and give the order | A ...Problem 21E:
21. Let
Be the special linear group of order over .Find the inverse of each of the following...Problem 22E:
22. Find the center for each of the following groups .
a. in Exercise 34 of section 3.1.
b. in...Problem 23E:
23. Let be the equivalence relation on defined by if and only if there exists an element in ...Problem 25E:
Let G be a group and Z(G) its center. Prove or disprove that if ab is in Z(G), then ab=ba.Problem 26E:
Let A be a given nonempty set. As noted in Example 2 of section 3.1, S(A) is a group with respect to...Problem 27E:
(See Exercise 26) Let A be an infinite set, and let H be the set of all fS(A) such that f(x)=x for...Problem 28E:
28. For each, define by for.
a. Show that is an element of .
b. Let .Prove that is a subgroup of ...Problem 29E:
Let G be an abelian group. For a fixed positive integer n, let Gn={ aGa=xnforsomexG }. Prove that Gn...Problem 30E:
For fixed integers a and b, let S={ ax+byxandy }. Prove that S is a subgroup of under addition.(A...Problem 31E:
31. a. Prove Theorem : The center of a group is an abelian subgroup of.
b. Prove Theorem :...Problem 32E:
Find the centralizer for each element a in each of the following groups. The quaternion group G={...Problem 38E:
Find subgroups H and K of the group S(A) in example 3 of section 3.1 such that HK is not a subgroup...Problem 39E:
39. Assume that and are subgroups of the abelian group. Prove that the set of products is a...Problem 40E:
40. Find subgroups and of the group in example of the section such that the set defined in...Problem 41E:
41. Let be a cyclic group, . Prove that is abelian.
Problem 42E:
Reword Definition 3.17 for an additive group G. Definition 3.17: Let G be a group. For any aG, the...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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