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Let
Find the distinct left cosets of
Find the distinct right cosets of
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Chapter 4 Solutions
ELEMENTS OF MODERN ALGEBRA
- Decide whether each of the following sets is a subgroup of G={ 1,1,i,i } under multiplication. If a set is not a subgroup, give a reason why is not. a. { 1,1 } b. { 1,i } c. { i,i } d. { 1,i }arrow_forwardLet be the subgroup of . 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 .arrow_forwardExercises 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_forward
- For each of the following subgroups H of the addition groups Z18, find the distinct left cosets of H in Z18, partition Z18 into left cosets of H, and state the index [ Z18:H ] of H in Z18. H= [ 8 ] .arrow_forwardLet 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?arrow_forward22. If and are both normal subgroups of , prove that is a normal subgroup of .arrow_forward
- 4. List all the elements of the subgroupin the group under addition, and state its order.arrow_forwardIn 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_forwardThe alternating group A4 on 4 elements is the same as the group D4 of symmetries for a square. That is. A4=D4.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage
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