In the multiplicative group
a.
b.
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Elements Of Modern Algebra
- Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.arrow_forward15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .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
- 13. Assume that are subgroups of the abelian group . Prove that if and only if is generated byarrow_forward5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:arrow_forwardLet be a subgroup of a group with . Prove that if and only if .arrow_forward
- Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.arrow_forward9. Find all homomorphic images of the octic group.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
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,