PROBLEM SET - Cyclic Groups1. Let Z denote the group of integers under addtion. Is everysubgroup of Z cyclic? Why? Describe all the subgroups of Z.2. List the elements of the (i), i.e. cyclic subgroup generated by iof the group C of nonzero complex numbers under multiplication.3. Let G = (a) and lal = 24. List all generators for the subgroup oforder 8.4. Draw the subgroup lattice diagram for Z36 and U(12).

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Answered: 1. Let Z denote the group of integers…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 27E: 27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. ...

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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?
The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is. A4=D4.
Exercises 10. Find an isomorphism from the multiplicative group to the group with multiplication table in Figure . This group is known as the Klein four group. Figure Sec. 16. a. Prove that each of the following sets is a subgroup of , the general linear group of order over . Sec. 3. Let be the Klein four group with its multiplication table given in Figure . Figure Sec. 17. Show that a group of order either is cyclic or is isomorphic to the Klein four group . Sec. 16. Repeat Exercise with the quaternion group , the Klein four group , and defined by
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 .
Find all Sylow 3-subgroups of the symmetric group S4.
Find all subgroups of the octic group D4.
34. Suppose that and are subgroups of the group . Prove that is a subgroup of .
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A subgroup H of the group Sn is called transitive on B=1,2,....,n if for each pair i,j of elements of B there exists an element hH such that h(i)=j. Suppose G is a group that is transitive on 1,2,....,n, and let Hi be the subgroup of G that leaves i fixed: Hi=gGg(i)=i For i=1,2,...,n. Prove that G=nHi.
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In Exercises 3 and 4, let be the octic group in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let be the subgroup of the octic group . Find the distinct left cosets of in , write out their elements, partition into left cosets of , and give . Find the distinct right cosets of in , write out their elements, and partition into right cosets of . Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group of rigid motions of a square The elements of the group are as follows: 1. the identity mapping 2. the counterclockwise rotation through about the center 3. the counterclockwise rotation through about the center 4. the counterclockwise rotation through about the center 5. the reflection about the horizontal line 6. the reflection about the diagonal 7. the reflection about the vertical line 8. the reflection about the diagonal . The dihedral group of rigid motions of the square is also known as the octic group. The multiplication table for is requested in Exercise 20 of this section.
In Exercises 3 and 4, let G be the octic group D4=e,,2,3,,,, in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. 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. Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group D4 of rigid motions of a square The elements of the group D4 are as follows: 1. the identity mapping e=(1) 2. the counterclockwise rotation =(1,2,3,4) through 900 about the center O 3. the counterclockwise rotation 2=(1,3)(2,4) through 1800 about the center O 4. the counterclockwise rotation 3=(1,4,3,2) through 2700 about the center O 5. the reflection =(1,4)(2,3) about the horizontal line h 6. the reflection =(2,4) about the diagonal d1 7. the reflection =(1,2)(3,4) about the vertical line v 8. the reflection =(1,3) about the diagonal d2. The dihedral group D4=e,,2,3,,,, of rigid motions of the square is also known as the octic group. The multiplication table for D4 is requested in Exercise 20 of this section.

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