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Suppose that G , G ' , and G ' ' are groups. If G ' is a homomorphic image of G , and G ' ' is a homomorphic image of G ' , prove that G ' ' is a homomorphic image of G . (Thus the relation in Exercise 15 has the transitive property.) Exercise 15: 15. Prove that on a given collection G of groups, the relation of being a homomorphic image has the reflexive property.

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Elements Of Modern Algebra

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

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Section
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230
Chapter 3.6, Problem 16E
Textbook Problem
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Suppose that G , G ' , and G ' ' are groups. If G ' is a homomorphic image of G , and G ' ' is a homomorphic image of G ' , prove that G ' ' is a homomorphic image of G . (Thus the relation in Exercise 15 has the transitive property.)

Exercise 15:

15. Prove that on a given collection G of groups, the relation of being a homomorphic image has the reflexive property.

To determine

To prove: G'' is homeomorphic image of G.

Explanation of Solution

Given information:

G,G' and G'' are groups. G' is homeomorphic image of G; G'' is homeomorphic image of G'.

Formula used:

1) Epimorphism: A homomorphism ϕ is called epimorphism if ϕ is onto.

2) Homomorphic Image: If there exists an epimorphism from G to G', then G' is called a homomorphic image.

Explanation:

Let ϕ,φ be a mapping from ϕ:GG' and φ:G'G''.

G' is homomorphic image of G, that is, ϕ is epimorphism.

G'' is homomorphic image of G', that is, φ is epimorphism.

Consider the composition of ϕ and φ is a mapping φϕ:GG''

To prove G'' is homomorphic image of G, it is sufficient to show that φϕ is epimorphism.

Let x,yG

φϕ(x*y)

By using composition of function,

=φ[ϕ(xy)]

As ϕ is homomorphism,

=φ[ϕ(x)*ϕ(x)]

φ is homomorphism

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Elements Of Modern Algebra
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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. 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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. 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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. 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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. 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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...

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