In Exercises 3 and 4, let
Let
Find the distinct left cosets of
Find the distinct right cosets of
Example 12
Using the notational convention described in the preceding paragraph, we shall write out the dihedral group
The elements of the group
1. the identity mapping
2. the counterclockwise rotation
3. the counterclockwise rotation
4. the counterclockwise rotation
5. the reflection
6. the reflection
7. the reflection
8. the reflection
The dihedral group
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Elements Of Modern Algebra
- 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.arrow_forwardIn Exercises 1- 9, let be the given group. Write out the elements of a group of permutations that is isomorphic to, and exhibit an isomorphism from to this group. 9. Let be 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 . 3. The quaternion group ; .arrow_forward
- In 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_forwardIn Exercises 1- 9, let G be the given group. Write out the elements of a group of permutations that is isomorphic to G, and exhibit an isomorphism from G to this group. Let G be the quaternion group { 1,i,j,k }arrow_forward
- 12. 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_forwardFind the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.arrow_forward31. (See Exercise 30.) Prove that if and are primes and is a nonabelian group of order , then the center of is the trivial subgroup . Exercise 30: 30. Let be a group with center . Prove that if is cyclic, then is abelian.arrow_forward
- Exercises 11. According to Exercise of section, if is prime, the nonzero elements of form a group with respect to multiplication. For each of the following values of , show that this group is cyclic. (Sec. ) a. b. c. d. e. f. 33. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.arrow_forwardIn Exercises 1- 9, let be the given group. Write out the elements of a group of permutations that is isomorphic to, and exhibit an isomorphism from to this group. 2. Let be the cyclic group of order.arrow_forwardLet be a subgroup of a group with . Prove that if and only if .arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,