Reading Question 5.3.1. Which of the following claims is true? Select all that apply.(a) ([3]10) is a proper subgroup of (Z10, +).(b) (3) is a proper subgroup of (Z,+).(c) ([3]10) = ([7]10), as subgroups of (Z10, +).(d) (3) = (7), as subgroups of (Z, +).

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Answered: Reading Question 5.3.1. Which of the…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.5: Normal Subgroups

Problem 15E: Find groups H and K such that the following conditions are satisfied: H is a normal subgroup of K. K...

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Let G be the group and H the subgroup given in each of the following exercises of Section 4.4. In each case, is H normal in G? Exercise 3 b. Exercise 4 c. Exercise 5 d. Exercise 6 e. Exercise 7 f. Exercise 8 Section 4.4 Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Let H be the subgroup (1),(2,3) of S3. Find the distinct left cosets of H in S3, write out their elements, partition S3 into left cosets of H, and give [S3:H]. Find the distinct right cosets of H in S3, write out their elements, and partition S3 into right cosets of H. In Exercises 7 and 8, let G be the multiplicative group of permutation matrices I3,P3,P32,P1,P4,P2 in Example 6 of Section 3.5 Let H be the subgroup of G given by H=I3,P4={ (100010001),(001010100) }. Find the distinct left cosets of H in G, write out their elements, partition G into left cosets of H, and give [G:H]. Find the distinct right cosets of H in G, write out their elements, and partition G into right cosets of H. Let H be the subgroup of G given by H=I3,P3,P32={ (100010001),(010001100),(001100010) }. Find the distinct left cosets of H in G, write out their elements, partition G into left cosets of H, and give [G:H]. Find the distinct right cosets of H in G, write out their elements, and partition G into right cosets of H.
Exercises 19. Find cyclic subgroups of that have three different orders.
Exercises In Section 3.3, the centralizer of an element a in the group G was shown to be the subgroup given by Ca=xGax=xa. Use the multiplication table constructed in Exercise 20 to find the centralizer Ca for each element a of the octic group D4. Construct a multiplication table for the octic group D4 described in Example 12 of this section.
9. Consider the octic group of Example 3. Find a subgroup of that has order and is a normal subgroup of . Find a subgroup of that has order and is not a normal subgroup of .
Find subgroups H and K of the group S(A) in example 3 of section 3.1 such that HK is not a subgroup of S(A). From Example 3 of section 3.1: A=1,2,3 and S(A) is a set of all permutations defined on A.
Exercises 1. List all cyclic subgroups of the group in Example of section. Example 3. We shall take and obtain an explicit example of . In order to define an element of , we need to specify , , and . There are three possible choices for . Since is to be bijective, there are two choices for after has been designated, and then only one choice for . Hence there are different mappings in .
Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?
5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:
Find groups H and K such that the following conditions are satisfied: H is a normal subgroup of K. K is a normal subgroup of the octic group. H is not a normal subgroup of the octic group.
True or False Label each of the following statements as either true or false. If a group G contains a normal subgroup, then every subgroup of G must be normal.
14. Find groups and such that and the following conditions are satisfied: a. is a normal subgroup of . b. is a normal subgroup of . c. is not a normal subgroup of . (Thus the statement “A normal subgroup of a normal subgroup is a normal subgroup” is false.)
27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.

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Reading Question 5.3.1. Which of the following claims is true? Select all that apply. (a) ([3]10) is a proper subgroup of (Z10,+). (b) (3) is a proper subgroup of (Z,+). (c) ([3]10) = ([7]10), as subgroups of (Z10, +). (d) (3) = (7), as subgroups of (Z,+).