For each of the following subgroups
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Chapter 4 Solutions
Elements Of Modern Algebra
- 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_forwardExercises 10. For each of the following values of, find all subgroups of the cyclic group under addition and state their order. a. b. c. d. e. f.arrow_forward6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.arrow_forward
- 1. Consider , the groups of units in under multiplication. For each of the following subgroups in , partition into left cosets of , and state the index of in a. b.arrow_forwardConsider the group U9 of all units in 9. Given that U9 is a cyclic group under multiplication, find all subgroups of U9.arrow_forward18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.arrow_forward
- Exercises 9. For each of the following values of, find all distinct generators of the cyclic group under addition. a. b. c. d. e. f.arrow_forwardExercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .arrow_forwardExercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.arrow_forward
- 11. Assume that are subgroups of the abelian group such that the sum is direct. If is a subgroup of for prove that is a direct sum.arrow_forwardExercises 8. Find an isomorphism from the group in Example of this section to the multiplicative group . Sec. 16. Prove that each of the following sets is a subgroup of , the general linear group of order over .arrow_forwardLet 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_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,