For the following problems, you will need the definition below.Definition: If N is a normal subgroup of a group G, then the cosets of N in G form a groupG/N under the operation (aN) (bN) = abN. This group is called the factor or quotient groupof G and N.(a) Show that H = {e, (1 2)} S3 but N = {e, (1 2 3), (3 2 1)} ◄ S3(b) Find S3/N and show that it is a group with the operation as defined above.(c) What group, of which we have seen many times now, is S3/N isomorphic to?(d) Consider the quotient group (Z4 × Z6)/((0,1)).

This question related to group theory.

Answered: For the following problems, you will…

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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Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.
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?
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 .
Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1 isomorphic to G2/H2. Determine the possible orders of H1 and H2 under the following conditions. a. G1=24 and G2=18 b. G1=32 and G2=40
Exercises 13. For each of the following values of, find all subgroups of the group described in Exercise, addition and state their order. a. b. c. d. e. f.
The alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is. A4=D4.
5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:
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.
Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .
Show that An has index 2 in Sn, and thereby conclude that An is always a normal subgroup of Sn.
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.
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.

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