Find an isomorphism from the group
Sec.
Prove that each of the following sets
Want to see the full answer?
Check out a sample textbook solutionChapter 3 Solutions
Elements Of Modern Algebra
- Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.arrow_forwardExercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .arrow_forward13. Assume that are subgroups of the abelian group . Prove that if and only if is generated byarrow_forward
- Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.arrow_forwardLet G be a group of finite order n. Prove that an=e for all a in G.arrow_forward5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:arrow_forward
- 18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.arrow_forward9. Find all homomorphic images of the octic group.arrow_forward11. 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_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_forwardProve part c of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.arrow_forward15. Prove that if for all in the group , then is abelian.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,