Let
Exercise 3 b. Exercise 4 c. Exercise 5
d. Exercise 6 e. Exercise 7 f. Exercise 8
Section 4.4
Let
Find the distinct left cosets of
Find the distinct right cosets of
Let
Find the distinct left cosets of
Find the distinct right cosets of
Let
Find the distinct left cosets of
Find the distinct right cosets of
Let
Find the distinct left cosets of
Find the distinct right cosets of
In Exercises 7 and 8, let
Let
Find the distinct left cosets of
Find the distinct right cosets of
Let
Find the distinct left cosets of
Find the distinct right cosets of
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Chapter 4 Solutions
Elements Of Modern Algebra
- 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.arrow_forwardIn 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.arrow_forwardFind the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.arrow_forward
- 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.arrow_forwardIn Exercises , is a normal subgroup of the group . Find the order of the quotient group . Write out the distinct elements of and construct a multiplication table for . 2. The octic group ; .arrow_forwardIn Exercises , is a normal subgroup of the group . Find the order of the quotient group . Write out the distinct elements of and construct a multiplication table for . 1. The octic group ; .arrow_forward
- Exercises 22. List all the distinct subgroups of each group in Exercise. Exercise 21. Suppose is a cyclic group of order. Determine the number of generators of for each value of and list all the distinct generators of . a. b. c. d. e. f.arrow_forwardFind all subgroups of the octic group D4.arrow_forward34. Suppose that and are subgroups of the group . Prove that is a subgroup of .arrow_forward
- Let be a group of order 24. If is a subgroup of , what are all the possible orders of ?arrow_forward18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.arrow_forward12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,