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8th Edition

Gilbert + 2 others

Publisher: Cengage Learning,

ISBN: 9781285463230

Chapter 3.6, Problem 16E

Textbook Problem

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Suppose that

**Exercise 15:**

**15.** Prove that on a given collection

To determine

**To prove:**

**Given information:**

**Formula used:**

1) Epimorphism: A homomorphism

2) Homomorphic Image: If there exists an epimorphism from

**Explanation:**

Let

Consider the composition of

To prove

Let

By using composition of function,

As

Elements Of Modern Algebra

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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 - Label each of the following statements as either...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 - 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 - In Exercises 114, decide whether each of the given...Ch. 3.1 - Exercises
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In Exercises, decide whether each of...Ch. 3.1 - Exercises
In Exercises, decide whether each of...Ch. 3.1 - Exercises
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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 - 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. 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 - 35. For an arbitrary in , the cyclic subgroup of...Ch. 3.3 - 36. Let , be an arbitrary nonempty collection of...Ch. 3.3 - 37. If is a group, prove that ,where is the...Ch. 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 - Exercises
18. Let be the group of units as...Ch. 3.4 - Exercises
19. 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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