G be defined by f(r) = x1. Prove that f is operation-preserving if 6*. Let G be a group and f: G and only if G is abelian
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- 9. Suppose that and are subgroups of the abelian group such that . Prove that .5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.
- Exercises 22. Let be a finite cyclic group of order with generators and . Prove that the mapping is an automorphism of .Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.16. Suppose that is an abelian group with respect to addition, with identity element Define a multiplication in by for all . Show that forms a ring with respect to these operations.